Subgroups — #30873215

Order 17152.

864 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14523999	C₂	1	1	yes	maximal, characteristic
2	#22560769	C₂	1	1	yes	maximal, characteristic, complemented
3	#22560769	C₂	1	1	yes	maximal, characteristic, complemented
4	#22560763	C₂	1	1	yes	maximal, characteristic, complemented
5	#22560763	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560389	C₂	1	1	yes	maximal, characteristic
7	#22560351	C₂	1	1	yes	maximal, characteristic, complemented
8	#22560325	C₂	1	1	yes	maximal, characteristic, complemented
9	#22560275	C₂	1	1	yes	maximal, characteristic
10	C₄⋊C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	#22559983	C₂	1	1	yes	maximal, characteristic
12	C₈×C₂²⋊Dic₆₇	C₂	1	1	yes	maximal, characteristic
13	#22559953	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#16863150	C₂	1	1	yes	maximal, characteristic
15	C₆₇×#1850	C₂	1	1	yes	maximal, characteristic
16	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×#14523999	C₄	1	1	yes	characteristic
18	C₂×#14523999	C₄	1	1	yes	characteristic
19	C₂²×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
23	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
27	C₄×Dic₆₇⋊C₄	C₂²	1	1	yes	characteristic
28	C₄×Dic₆₇⋊C₄	C₂²	1	1	yes	characteristic
29	#16863247	C₄	1	1	yes	characteristic
30	#16863247	C₄	1	1	yes	characteristic
31	C₂²⋊C₄×Dic₆₇	C₄	1	1	yes	characteristic
32	C₂²⋊C₄×Dic₆₇	C₄	1	1	yes	characteristic
33	C₄×C₂²⋊Dic₆₇	C₂²	1	1	yes	characteristic
34	#16863241	C₄	1	1	yes	characteristic
35	#16863241	C₄	1	1	yes	characteristic
36	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
37	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
38	#16863151	C₂²	1	1	yes	characteristic
39	#16863151	C₂²	1	1	yes	characteristic
40	#16863150	C₂²	1	1	yes	characteristic
41	#16863150	C₂²	1	1	yes	characteristic
42	#16863150	C₂²	1	1	yes	characteristic
43	#16863150	C₂²	1	1	yes	characteristic
44	#16863150	C₂²	1	1	yes	characteristic
45	#16863150	C₂²	1	1	yes	characteristic
46	#16863150	C₂²	1	1	yes	characteristic
47	#16863150	C₂²	1	1	yes	characteristic
48	#16863150	C₂²	1	1	yes	characteristic
49	#16863150	C₂²	1	1	yes	characteristic
50	#16863150	C₂²	1	1	yes	characteristic
51	#16863150	C₂²	1	1	yes	characteristic
52	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
53	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
54	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
55	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
56	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
57	#336×C₆₇	C₂²	1	1	yes	characteristic
58	#336×C₆₇	C₂²	1	1	yes	characteristic
59	C₂³×Dic₆₇	C₂×C₄	1	1	yes	characteristic
60	C₂³×Dic₆₇	C₂×C₄	1	1	yes	characteristic
61	C₂²×C₆₇⋊C₈	—	2	2	no	—
62	C₂²×C₆₇⋊C₈	—	2	2	no	—
63	C₂²×C₆₇⋊C₈	—	4	1	no	—
64	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
65	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
66	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
67	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
68	#14523999	—	2	4	no	—
69	#14523999	C₈	1	8	yes	complemented
70	C₂×Dic₆₇⋊C₄	C₂×C₄	1	1	yes	characteristic
71	C₂×Dic₆₇⋊C₄	C₂×C₄	1	1	yes	characteristic
72	C₂×Dic₆₇⋊C₄	C₂×C₄	1	1	yes	characteristic
73	C₂×Dic₆₇⋊C₄	C₂×C₄	1	1	yes	characteristic
74	C₂×C₂²⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
75	C₂×C₂²⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄×Dic₆₇	—	2	2	no	—
77	C₂×C₄×Dic₆₇	—	2	2	no	—
78	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
79	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
80	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
81	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
82	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
83	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
84	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
85	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
86	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
87	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
88	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
89	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
90	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
91	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
92	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
93	C₄⋊(C₆₇⋊C₈)	—	4	1	no	—
94	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
95	C₈×Dic₆₇	—	2	2	no	—
96	C₈×Dic₆₇	—	2	2	no	—
97	C₈×Dic₆₇	—	4	1	no	—
98	C₈×Dic₆₇	—	4	1	no	—
99	C₆₇⋊#95	C₂×C₄	1	1	yes	characteristic
100	C₆₇⋊#95	C₂×C₄	1	1	yes	characteristic
101	C₆₇⋊#95	C₂×C₄	1	1	yes	characteristic
102	C₆₇⋊#95	C₂×C₄	1	1	yes	characteristic
103	C₆₇⋊#95	C₂×C₄	1	1	yes	characteristic
104	C₆₇⋊#95	C₂×C₄	1	1	yes	characteristic
105	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
106	C₂²×C₈×C₆₇	—	2	2	no	—
107	C₂²×C₈×C₆₇	—	2	2	no	—
108	C₂²×C₈×C₆₇	—	4	1	no	—
109	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
110	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
111	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
112	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
113	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
114	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
115	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
116	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
117	C₄⋊C₈×C₆₇	—	2	2	no	—
118	C₄⋊C₈×C₆₇	—	4	1	no	—
119	C₂²⋊C₈×C₆₇	—	2	2	no	—
120	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
121	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
122	C₂²×Dic₆₇	—	2	2	no	—
123	C₂²×Dic₆₇	—	2	2	no	—
124	C₂²×Dic₆₇	—	2	2	no	—
125	C₂²×Dic₆₇	—	2	2	no	—
126	C₂²×Dic₆₇	—	4	1	no	—
127	C₂²×Dic₆₇	C₂×C₈	1	4	yes	—
128	C₂²×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
129	C₂²×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
130	C₂²×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
131	C₂²×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
132	C₂²×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
133	C₂²×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
134	C₂²×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
135	C₂²×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
136	C₂×C₆₇⋊C₈	—	2	1	no	—
137	C₂×C₆₇⋊C₈	—	2	1	no	—
138	C₂×C₆₇⋊C₈	—	2	1	no	—
139	C₂×C₆₇⋊C₈	—	2	1	no	—
140	C₂×C₆₇⋊C₈	—	2	1	no	—
141	C₂×C₆₇⋊C₈	—	2	1	no	—
142	C₂×C₆₇⋊C₈	—	2	1	no	—
143	C₂×C₆₇⋊C₈	—	2	1	no	—
144	C₂×C₆₇⋊C₈	—	2	1	no	—
145	C₂×C₆₇⋊C₈	—	2	1	no	—
146	C₂×C₆₇⋊C₈	—	2	1	no	—
147	C₂×C₆₇⋊C₈	—	4	4	no	—
148	C₂×C₆₇⋊C₈	#41	1	2	yes	—
149	Dic₆₇⋊C₄	—	2	2	no	—
150	Dic₆₇⋊C₄	—	2	2	no	—
151	Dic₆₇⋊C₄	C₂×C₈	1	4	yes	complemented
152	Dic₆₇⋊C₄	C₂×C₈	1	4	yes	complemented
153	C₂²⋊Dic₆₇	C₂×C₈	1	4	yes	—
154	C₂²⋊Dic₆₇	M₁₆	1	4	yes	—
155	C₄×Dic₆₇	—	2	2	no	—
156	C₄×Dic₆₇	—	2	2	no	—
157	C₄×Dic₆₇	—	2	1	no	—
158	C₄×Dic₆₇	—	2	1	no	—
159	C₄×Dic₆₇	—	2	1	no	—
160	C₄×Dic₆₇	—	2	1	no	—
161	C₄×Dic₆₇	—	2	1	no	—
162	C₄×Dic₆₇	—	2	1	no	—
163	C₄×Dic₆₇	—	2	1	no	—
164	C₄×Dic₆₇	—	2	1	no	—
165	C₄×Dic₆₇	—	2	2	no	—
166	C₄×Dic₆₇	—	2	2	no	—
167	C₄×Dic₆₇	—	2	2	no	—
168	C₄×Dic₆₇	—	2	2	no	—
169	C₄×Dic₆₇	—	4	1	no	—
170	C₄×Dic₆₇	—	4	1	no	—
171	C₄×Dic₆₇	C₂×C₈	1	4	yes	complemented
172	C₄×Dic₆₇	C₂×C₈	1	4	yes	complemented
173	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
174	C₂²×C₄×C₆₇	—	2	1	no	—
175	C₂²×C₄×C₆₇	—	2	1	no	—
176	C₂²×C₄×C₆₇	—	2	1	no	—
177	C₂²×C₄×C₆₇	—	2	1	no	—
178	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
179	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
180	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
181	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
182	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
183	C₂²×C₄×C₆₇	#41	1	2	yes	—
184	C₂²×C₄×C₆₇	#41	1	2	yes	—
185	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
186	C₂×C₈×C₆₇	—	2	1	no	—
187	C₂×C₈×C₆₇	—	2	1	no	—
188	C₂×C₈×C₆₇	—	2	1	no	—
189	C₂×C₈×C₆₇	—	2	1	no	—
190	C₂×C₈×C₆₇	—	2	1	no	—
191	C₂×C₈×C₆₇	—	2	1	no	—
192	C₂×C₈×C₆₇	—	2	1	no	—
193	C₂×C₈×C₆₇	—	2	1	no	—
194	C₂×C₈×C₆₇	—	2	1	no	—
195	C₂×C₈×C₆₇	—	2	1	no	—
196	C₂×C₈×C₆₇	—	2	1	no	—
197	C₂×C₈×C₆₇	—	4	4	no	—
198	C₂×C₈×C₆₇	#41	1	2	yes	—
199	C₂²⋊C₄×C₆₇	—	2	2	no	—
200	C₂²⋊C₄×C₆₇	C₂×C₈	1	4	yes	—
201	C₄²×C₆₇	—	2	1	no	—
202	C₄²×C₆₇	—	2	1	no	—
203	C₄²×C₆₇	—	2	1	no	—
204	C₄²×C₆₇	—	2	1	no	—
205	C₂³×C₆₇	—	2	1	no	—
206	C₂³×C₆₇	—	2	1	no	—
207	C₂³×C₆₇	—	2	1	no	—
208	C₂³×C₆₇	—	2	1	no	—
209	C₂³×C₆₇	#117	1	2	yes	—
210	C₂³×C₆₇	C₂²×C₈	1	2	yes	—
211	C₂³×C₆₇	C₂×M₁₆	1	2	yes	—
212	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
213	C₂×C₄×C₆₇	—	2	1	no	—
214	C₂×C₄×C₆₇	—	2	1	no	—
215	C₂×C₄×C₆₇	—	2	1	no	—
216	C₂×C₄×C₆₇	—	2	1	no	—
217	C₂×C₄×C₆₇	—	2	1	no	—
218	C₂×C₄×C₆₇	—	2	1	no	—
219	C₂×C₄×C₆₇	—	2	2	no	—
220	C₂×C₄×C₆₇	—	2	2	no	—
221	C₂×C₄×C₆₇	—	2	2	no	—
222	C₂×C₄×C₆₇	—	2	2	no	—
223	C₂×C₄×C₆₇	—	2	1	no	—
224	C₂×C₄×C₆₇	—	2	1	no	—
225	C₂×C₄×C₆₇	—	2	1	no	—
226	C₂×C₄×C₆₇	—	2	1	no	—
227	C₂×C₄×C₆₇	—	2	1	no	—
228	C₂×C₄×C₆₇	—	2	1	no	—
229	C₂×C₄×C₆₇	#117	1	2	yes	—
230	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
231	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
232	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
233	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
234	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
235	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
236	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
237	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
238	C₂×Dic₆₇	—	2	1	no	—
239	C₂×Dic₆₇	—	2	1	no	—
240	C₂×Dic₆₇	—	2	1	no	—
241	C₂×Dic₆₇	—	2	1	no	—
242	C₂×Dic₆₇	—	2	1	no	—
243	C₂×Dic₆₇	—	2	1	no	—
244	C₂×Dic₆₇	—	2	1	no	—
245	C₂×Dic₆₇	—	2	1	no	—
246	C₂×Dic₆₇	—	2	1	no	—
247	C₂×Dic₆₇	—	2	1	no	—
248	C₂×Dic₆₇	—	2	1	no	—
249	C₂×Dic₆₇	—	2	1	no	—
250	C₂×Dic₆₇	—	2	1	no	—
251	C₂×Dic₆₇	—	2	1	no	—
252	C₂×Dic₆₇	—	2	8	no	—
253	C₂×Dic₆₇	—	2	1	no	—
254	C₂×Dic₆₇	—	2	1	no	—
255	C₂×Dic₆₇	—	2	1	no	—
256	C₂×Dic₆₇	—	4	4	no	—
257	C₂×Dic₆₇	C₂²×C₈	1	2	yes	—
258	C₂×Dic₆₇	C₂²×C₈	1	2	yes	—
259	C₂×Dic₆₇	C₂²×C₈	1	2	yes	—
260	C₂×Dic₆₇	C₂²×C₈	1	2	yes	—
261	C₂×Dic₆₇	C₂×M₁₆	1	2	yes	—
262	C₂×Dic₆₇	C₂×M₁₆	1	2	yes	—
263	C₂×Dic₆₇	#131	1	2	yes	—
264	C₈×C₆₇	—	2	2	no	—
265	C₈×C₆₇	—	4	1	no	—
266	C₈×C₆₇	—	4	1	no	—
267	C₈×C₆₇	—	4	1	no	—
268	C₆₇⋊C₈	—	2	2	no	—
269	C₆₇⋊C₈	—	4	1	no	—
270	C₆₇⋊C₈	—	4	1	no	—
271	C₆₇⋊C₈	—	4	1	no	—
272	C₂²×C₆₇	—	2	2	no	—
273	C₂²×C₆₇	—	2	2	no	—
274	C₂²×C₆₇	—	2	2	no	—
275	C₂²×C₆₇	—	2	2	no	—
276	C₂²×C₆₇	—	2	2	no	—
277	C₂²×C₆₇	—	2	2	no	—
278	C₂²×C₆₇	#431	1	4	yes	—
279	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
280	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
281	C₂²×C₆₇	#520	1	1	yes	characteristic
282	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
283	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
284	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
285	C₂²×C₆₇	#568	1	1	yes	characteristic
286	C₄×C₆₇	—	2	2	no	—
287	C₄×C₆₇	—	2	2	no	—
288	C₄×C₆₇	—	2	2	no	—
289	C₄×C₆₇	—	4	1	no	—
290	C₄×C₆₇	—	4	1	no	—
291	C₄×C₆₇	#551	1	1	yes	characteristic
292	C₄×C₆₇	#551	1	1	yes	characteristic
293	C₄×C₆₇	#552	1	1	yes	characteristic
294	C₄×C₆₇	#552	1	1	yes	characteristic
295	Dic₆₇	—	2	2	no	—
296	Dic₆₇	—	2	2	no	—
297	Dic₆₇	—	2	2	no	—
298	Dic₆₇	—	2	2	no	—
299	Dic₆₇	—	2	2	no	—
300	Dic₆₇	—	2	2	no	—
301	Dic₆₇	—	2	2	no	—
302	Dic₆₇	—	4	1	no	—
303	#22589	—	67	1	no	maximal, Sylow
304	C₂×C₆₇	—	2	4	no	—
305	C₂×C₆₇	#2343	1	1	yes	characteristic
306	C₂×C₆₇	C₂×#431	1	1	yes	characteristic
307	C₂×C₆₇	#2924	1	1	yes	characteristic
308	C₂×C₆₇	C₈×#41	1	1	yes	characteristic
309	C₂×C₆₇	#2970	1	1	yes	characteristic
310	C₂×C₆₇	#2988	1	1	yes	characteristic
311	C₂×C₆₇	#2990	1	1	yes	characteristic
312	C₄×#117	—	67	1	no	—
313	#1916	—	67	1	no	—
314	#1916	—	67	1	no	—
315	#1916	—	67	1	no	—
316	#1916	—	67	1	no	—
317	#1850	—	67	1	no	—
318	#1850	D₆₇	1	1	yes	characteristic
319	#1839	—	67	1	no	—
320	#1831	—	67	1	no	—
321	#1817	—	67	1	no	—
322	C₄×C₄⋊C₈	—	67	1	no	—
323	C₈×C₂²⋊C₄	—	67	1	no	—
324	C₈×C₂²⋊C₄	—	67	1	no	—
325	#1744	—	67	1	no	—
326	C₂×#336	—	67	1	no	—
327	C₆₇	#22589	1	1	yes	characteristic, complemented, Sylow
328	C₂³×C₈	—	67	1	no	—
329	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
330	C₂×#117	—	67	1	no	—
331	C₂×#117	—	67	1	no	—
332	C₂²×C₄²	—	67	1	no	—
333	C₂×C₄⋊C₈	—	67	1	no	—
334	C₂×C₄⋊C₈	—	67	1	no	—
335	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
336	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
337	C₂×C₂²⋊C₈	—	67	1	no	—
338	C₂×C₂²⋊C₈	C₂×D₆₇	1	1	yes	characteristic
339	C₂×C₄×C₈	—	67	1	no	—
340	C₂×C₄×C₈	—	67	1	no	—
341	C₂×C₄×C₈	—	67	1	no	—
342	C₂×C₄×C₈	—	67	1	no	—
343	C₄×C₄⋊C₄	—	67	1	no	—
344	C₄×C₄⋊C₄	—	67	1	no	—
345	C₄×C₄⋊C₄	—	67	1	no	—
346	C₄×C₄⋊C₄	—	67	1	no	—
347	C₄×C₂²⋊C₄	—	67	1	no	—
348	C₄×C₂²⋊C₄	—	67	1	no	—
349	C₄×C₂²⋊C₄	—	67	1	no	—
350	C₄×C₂²⋊C₄	C₂×D₆₇	1	1	yes	characteristic
351	#376	—	67	1	no	—
352	#376	—	67	1	no	—
353	C₄³	—	67	1	no	—
354	C₄³	—	67	1	no	—
355	#336	—	67	1	no	—
356	#336	—	67	1	no	—
357	#336	—	67	1	no	—
358	#336	—	67	1	no	—
359	#336	—	67	1	no	—
360	#336	—	67	1	no	—
361	#336	—	67	1	no	—
362	#336	—	67	1	no	—
363	#336	—	67	1	no	—
364	#336	—	67	1	no	—
365	#336	—	67	1	no	—
366	#336	—	67	1	no	—
367	#336	—	67	1	no	—
368	#336	—	67	1	no	—
369	#336	C₂×D₆₇	1	1	yes	characteristic
370	#336	C₂×D₆₇	1	1	yes	characteristic
371	C₂³×C₄	—	67	1	no	—
372	C₂³×C₄	—	67	1	no	—
373	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
374	C₂²×C₈	—	2	2	no	—
375	C₂²×C₈	—	2	2	no	—
376	C₂²×C₈	—	4	1	no	—
377	C₂²×C₈	—	268	1	no	—
378	C₂²×C₈	—	67	1	no	—
379	C₂²×C₈	—	67	1	no	—
380	C₂²×C₈	—	67	1	no	—
381	C₂²×C₈	—	67	1	no	—
382	C₂²×C₈	—	134	2	no	—
383	C₂²×C₈	—	134	2	no	—
384	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
385	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
386	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
387	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
388	#117	—	67	8	no	—
389	#117	—	134	4	no	—
390	C₂×C₄⋊C₄	—	67	1	no	—
391	C₂×C₄⋊C₄	—	67	1	no	—
392	C₂×C₄⋊C₄	—	67	1	no	—
393	C₂×C₄⋊C₄	—	67	1	no	—
394	C₂×C₂²⋊C₄	—	67	1	no	—
395	C₂×C₂²⋊C₄	—	67	1	no	—
396	C₂×C₂²⋊C₄	C₄×D₆₇	1	1	yes	characteristic
397	C₂×C₂²⋊C₄	C₄×D₆₇	1	1	yes	characteristic
398	C₂×C₄²	—	67	1	no	—
399	C₂×C₄²	—	67	1	no	—
400	C₂×C₄²	—	67	1	no	—
401	C₂×C₄²	—	67	1	no	—
402	C₂×C₄²	—	67	1	no	—
403	C₂×C₄²	—	67	1	no	—
404	C₂×C₄²	—	67	1	no	—
405	C₂×C₄²	—	67	1	no	—
406	C₂×C₄²	—	67	1	no	—
407	C₂×C₄²	—	67	1	no	—
408	C₂×C₄²	—	67	1	no	—
409	C₂×C₄²	—	67	1	no	—
410	C₂×C₄²	—	67	1	no	—
411	C₂×C₄²	—	67	1	no	—
412	C₂×C₄²	—	134	2	no	—
413	C₂×C₄²	—	134	2	no	—
414	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
415	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
416	C₄⋊C₈	—	2	2	no	—
417	C₄⋊C₈	—	4	1	no	—
418	C₄⋊C₈	—	268	1	no	—
419	C₄⋊C₈	—	134	2	no	—
420	C₂²⋊C₈	—	2	2	no	—
421	C₂²⋊C₈	—	134	2	no	—
422	C₄×C₈	—	268	1	no	—
423	C₄×C₈	—	268	1	no	—
424	C₄×C₈	—	134	2	no	—
425	C₄×C₈	—	134	2	no	—
426	#95	—	67	1	no	—
427	#95	—	67	1	no	—
428	#95	—	67	1	no	—
429	#95	—	67	1	no	—
430	#95	—	67	1	no	—
431	#95	—	67	1	no	—
432	#95	C₄×D₆₇	1	1	yes	characteristic
433	#95	C₄×D₆₇	1	1	yes	characteristic
434	C₂⁴	C₂×C₄×D₆₇	1	1	yes	characteristic
435	C₂²×C₄	—	2	1	no	—
436	C₂²×C₄	—	2	1	no	—
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	—	268	1	no	—
440	C₂²×C₄	—	67	4	no	—
441	C₂²×C₄	—	67	1	no	—
442	C₂²×C₄	—	67	1	no	—
443	C₂²×C₄	—	67	1	no	—
444	C₂²×C₄	—	67	1	no	—
445	C₂²×C₄	—	67	1	no	—
446	C₂²×C₄	—	67	1	no	—
447	C₂²×C₄	—	67	1	no	—
448	C₂²×C₄	—	67	1	no	—
449	C₂²×C₄	—	134	2	no	—
450	C₂²×C₄	—	134	2	no	—
451	C₂²×C₄	—	134	2	no	—
452	C₂²×C₄	—	134	2	no	—
453	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
454	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
455	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
456	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
457	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
458	C₂²×C₄	#11761627	1	2	yes	—
459	C₂²×C₄	#11761629	1	2	yes	—
460	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
461	C₂×C₈	—	2	1	no	—
462	C₂×C₈	—	2	1	no	—
463	C₂×C₈	—	2	1	no	—
464	C₂×C₈	—	2	1	no	—
465	C₂×C₈	—	2	1	no	—
466	C₂×C₈	—	2	1	no	—
467	C₂×C₈	—	2	1	no	—
468	C₂×C₈	—	2	1	no	—
469	C₂×C₈	—	2	1	no	—
470	C₂×C₈	—	2	1	no	—
471	C₂×C₈	—	2	1	no	—
472	C₂×C₈	—	4	4	no	—
473	C₂×C₈	—	268	4	no	—
474	C₂×C₈	—	67	2	no	—
475	C₂×C₈	—	134	1	no	—
476	C₂×C₈	—	134	1	no	—
477	C₂×C₈	—	134	1	no	—
478	C₂×C₈	—	134	1	no	—
479	C₂×C₈	—	134	1	no	—
480	C₂×C₈	—	134	1	no	—
481	C₂×C₈	—	134	1	no	—
482	C₂×C₈	—	134	1	no	—
483	C₂×C₈	—	134	1	no	—
484	C₂×C₈	—	134	1	no	—
485	C₂×C₈	—	134	1	no	—
486	C₂×C₈	#11761627	1	2	yes	—
487	C₄⋊C₄	—	67	4	no	—
488	C₄⋊C₄	—	67	4	no	—
489	C₄⋊C₄	—	134	2	no	—
490	C₄⋊C₄	—	134	2	no	—
491	C₂²⋊C₄	—	2	2	no	—
492	C₂²⋊C₄	—	67	4	no	—
493	C₂²⋊C₄	—	67	4	no	—
494	C₂²⋊C₄	C₈×D₆₇	1	4	yes	—
495	C₄²	—	2	1	no	—
496	C₄²	—	2	1	no	—
497	C₄²	—	2	1	no	—
498	C₄²	—	2	1	no	—
499	C₄²	—	268	1	no	—
500	C₄²	—	268	1	no	—
501	C₄²	—	67	4	no	—
502	C₄²	—	67	4	no	—
503	C₄²	—	134	2	no	—
504	C₄²	—	134	2	no	—
505	C₄²	—	134	2	no	—
506	C₄²	—	134	2	no	—
507	C₄²	—	134	2	no	—
508	C₄²	—	134	2	no	—
509	C₄²	—	134	1	no	—
510	C₄²	—	134	1	no	—
511	C₄²	—	134	1	no	—
512	C₄²	—	134	1	no	—
513	C₄²	—	134	1	no	—
514	C₄²	—	134	1	no	—
515	C₄²	—	134	1	no	—
516	C₄²	—	134	1	no	—
517	C₂³	—	2	1	no	—
518	C₂³	—	2	1	no	—
519	C₂³	—	2	1	no	—
520	C₂³	—	2	1	no	—
521	C₂³	#14524001	1	2	yes	—
522	C₂³	C₂×C₈×D₆₇	1	2	yes	—
523	C₂³	M₁₆×D₆₇	1	2	yes	—
524	C₂³	C₂²×C₄×D₆₇	1	1	yes	characteristic
525	C₂×C₄	—	2	1	no	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	2	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	1	no	—
539	C₂×C₄	—	2	1	no	—
540	C₂×C₄	—	2	1	no	—
541	C₂×C₄	—	268	4	no	—
542	C₂×C₄	—	67	2	no	—
543	C₂×C₄	—	67	2	no	—
544	C₂×C₄	—	67	2	no	—
545	C₂×C₄	—	67	2	no	—
546	C₂×C₄	—	67	2	no	—
547	C₂×C₄	—	67	2	no	—
548	C₂×C₄	—	67	2	no	—
549	C₂×C₄	—	134	1	no	—
550	C₂×C₄	—	134	1	no	—
551	C₂×C₄	—	134	1	no	—
552	C₂×C₄	—	134	1	no	—
553	C₂×C₄	—	134	1	no	—
554	C₂×C₄	—	134	1	no	—
555	C₂×C₄	—	134	1	no	—
556	C₂×C₄	—	134	1	no	—
557	C₂×C₄	—	134	1	no	—
558	C₂×C₄	—	134	1	no	—
559	C₂×C₄	—	134	1	no	—
560	C₂×C₄	—	134	1	no	—
561	C₂×C₄	—	134	1	no	—
562	C₂×C₄	—	134	1	no	—
563	C₂×C₄	—	134	1	no	—
564	C₂×C₄	—	134	1	no	—
565	C₂×C₄	—	134	1	no	—
566	C₂×C₄	—	134	8	no	—
567	C₂×C₄	#14524001	1	2	yes	—
568	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
569	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
570	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
571	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
572	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
573	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
574	C₂×C₄	#14524088	1	1	yes	characteristic
575	C₂×C₄	#14524090	1	1	yes	characteristic
576	C₈	—	2	2	no	—
577	C₈	—	4	1	no	—
578	C₈	—	4	1	no	—
579	C₈	—	4	1	no	—
580	C₈	—	268	1	no	—
581	C₈	—	268	1	no	—
582	C₈	—	268	1	no	—
583	C₈	—	134	2	no	—
584	C₂²	—	2	2	no	—
585	C₂²	—	2	2	no	—
586	C₂²	—	2	2	no	—
587	C₂²	—	2	2	no	—
588	C₂²	—	2	2	no	—
589	C₂²	—	2	2	no	—
590	C₂²	#16863448	1	4	yes	—
591	C₂²	C₂×#14524001	1	1	yes	characteristic
592	C₂²	C₄×#11761627	1	1	yes	characteristic
593	C₂²	#16863875	1	1	yes	characteristic
594	C₂²	C₂²×C₈×D₆₇	1	1	yes	characteristic
595	C₂²	C₂×M₁₆×D₆₇	1	1	yes	characteristic
596	C₂²	#131×D₆₇	1	1	yes	characteristic
597	C₂²	#16864161	1	1	yes	characteristic
598	C₄	—	2	2	no	—
599	C₄	—	2	2	no	—
600	C₄	—	2	2	no	—
601	C₄	—	4	1	no	—
602	C₄	—	4	1	no	—
603	C₄	—	268	1	no	—
604	C₄	—	134	2	no	—
605	C₄	—	134	2	no	—
606	C₄	—	134	2	no	—
607	C₄	—	134	2	no	—
608	C₄	—	134	2	no	—
609	C₄	—	134	2	no	—
610	C₄	—	134	2	no	—
611	C₄	#16864073	1	1	yes	characteristic
612	C₄	#16864077	1	1	yes	characteristic
613	C₄	#16864083	1	1	yes	characteristic
614	C₄	#16864088	1	1	yes	characteristic
615	C₂	—	2	4	no	—
616	C₂	#22563067	1	1	yes	characteristic
617	C₂	C₂×#16863448	1	1	yes	characteristic
618	C₂	#22568866	1	1	yes	characteristic
619	C₂	C₈×#11761627	1	1	yes	characteristic
620	C₂	#22569219	1	1	yes	characteristic
621	C₂	#22569316	1	1	yes	characteristic
622	C₂	#22569329	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator	C₂²×C₆₇	C₂³×C₈
Fitting	C₆₇×#1850	C₂
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂⁴×C₆₇	C₂²×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1850	D₆₇
hypocenter	C₆₇	#22589
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ #1850

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #22560769	C₂	semidirect
3. #22560769	C₂	semidirect
4. #22560763	C₂	semidirect
5. #22560763	C₂	semidirect
7. #22560351	C₂	semidirect
8. #22560325	C₂	semidirect
10. C₄⋊C₈×Dic₆₇	C₂	semidirect
13. #22559953	C₂	semidirect
69. #14523999	C₈	semidirect
151. Dic₆₇⋊C₄	C₂×C₈	semidirect
152. Dic₆₇⋊C₄	C₂×C₈	semidirect
171. C₄×Dic₆₇	C₂×C₈	semidirect
172. C₄×Dic₆₇	C₂×C₈	semidirect
327. C₆₇	#22589	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×#14523999	2
#22560769	2
#22560763	2
#22560389	2
#22560351	2
#22560325	2
#22560275	2
C₄⋊C₈×Dic₆₇	2
#22559983	2
C₈×C₂²⋊Dic₆₇	2
#22559953	2
C₂×#16863150	2
C₆₇×#1850	2
#22589	67