Subgroups — #30866341

Order 17152.

774 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863458	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863421	C₂	1	1	yes	maximal, characteristic
3	#22562702	C₂	1	1	yes	maximal, characteristic
4	#22560932	C₂	1	1	yes	maximal, characteristic, complemented
5	#22560928	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560925	C₂	1	1	yes	maximal, characteristic, complemented
7	#22560393	C₂	1	1	yes	maximal, characteristic
8	#22560390	C₂	1	1	yes	maximal, characteristic
9	#22560388	C₂	1	1	yes	maximal, characteristic
10	#22560385	C₂	1	1	yes	maximal, characteristic
11	#22560362	C₂	1	1	yes	maximal, characteristic, complemented
12	#22560360	C₂	1	1	yes	maximal, characteristic, complemented
13	#22560327	C₂	1	1	yes	maximal, characteristic, complemented
14	#22560324	C₂	1	1	yes	maximal, characteristic, complemented
15	C₆₇×#1940	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×#11761611	C₂²	1	1	yes	characteristic
17	C₂²×C₈×D₆₇	C₂²	1	1	yes	characteristic
18	C₂×#14524000	C₂²	1	1	yes	characteristic
19	#16863458	—	2	4	no	—
20	#16863421	—	2	4	no	—
21	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
24	C₂×Dic₆₇⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×Dic₆₇⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×Dic₆₇⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×Dic₆₇⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×Dic₆₇⋊C₈	C₂²	1	1	yes	characteristic
29	C₂×Dic₆₇⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×Dic₆₇⋊C₈	C₂²	1	1	yes	characteristic
31	C₂×Dic₆₇⋊C₈	C₂²	1	1	yes	characteristic
32	C₂×(C₂×D₆₇)⋊C₈	C₂²	1	1	yes	characteristic
33	C₂×(C₂×D₆₇)⋊C₈	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
35	#16863288	C₄	1	1	yes	characteristic
36	#16863288	C₄	1	1	yes	characteristic
37	#16863286	C₄	1	1	yes	characteristic
38	#16863286	C₄	1	1	yes	characteristic
39	#16863283	C₄	1	1	yes	characteristic
40	#16863283	C₄	1	1	yes	characteristic
41	#16863277	C₄	1	1	yes	characteristic
42	#16863277	C₄	1	1	yes	characteristic
43	C₄×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
44	C₄×(C₂×D₆₇)⋊C₄	C₂²	1	1	yes	characteristic
45	C₄×(C₂×D₆₇)⋊C₄	C₂²	1	1	yes	characteristic
46	#16863242	C₂²	1	1	yes	characteristic
47	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
48	#16863151	C₂²	1	1	yes	characteristic
49	#16863151	C₂²	1	1	yes	characteristic
50	#16863150	C₂²	1	1	yes	characteristic
51	#16863150	C₂²	1	1	yes	characteristic
52	#16863150	C₂²	1	1	yes	characteristic
53	#16863150	C₂²	1	1	yes	characteristic
54	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
55	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
56	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
57	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
58	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
59	#336×C₆₇	C₂²	1	1	yes	characteristic
60	#336×C₆₇	C₂²	1	1	yes	characteristic
61	C₂²×C₄×D₆₇	C₂³	1	1	yes	characteristic
62	C₂×#11761611	—	2	2	no	—
63	C₂×#11761611	—	4	1	no	—
64	C₂×#11761611	—	4	1	no	—
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	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
69	C₂×C₈×D₆₇	—	2	2	no	—
70	C₂×C₈×D₆₇	—	4	1	no	—
71	C₂×C₈×D₆₇	—	4	1	no	—
72	#14524000	D₄	1	4	yes	—
73	#14524000	Q₈	1	4	yes	—
74	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
76	C₂×Dic₆₇⋊C₄	C₂×C₄	1	1	yes	characteristic
77	C₂×Dic₆₇⋊C₄	C₂×C₄	1	1	yes	characteristic
78	C₂×(C₂×D₆₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
79	C₂×(C₂×D₆₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
80	C₂×(C₂×D₆₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
81	C₂×(C₂×D₆₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
82	C₂×(C₂×D₆₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
83	C₂×(C₂×D₆₇)⋊C₄	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	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
92	C₄⋊(C₆₇⋊C₈)	—	4	1	no	—
93	C₄⋊(C₆₇⋊C₈)	—	4	1	no	—
94	Dic₆₇⋊C₈	—	2	2	no	—
95	Dic₆₇⋊C₈	—	2	2	no	—
96	Dic₆₇⋊C₈	—	2	2	no	—
97	Dic₆₇⋊C₈	—	2	2	no	—
98	Dic₆₇⋊C₈	—	4	1	no	—
99	Dic₆₇⋊C₈	—	4	1	no	—
100	Dic₆₇⋊C₈	—	4	1	no	—
101	Dic₆₇⋊C₈	—	4	1	no	—
102	(C₂×D₆₇)⋊C₈	—	2	2	no	—
103	(C₂×D₆₇)⋊C₈	—	2	2	no	—
104	C₄×C₆₇⋊C₈	—	2	2	no	—
105	C₆₇⋊#95	C₂×C₄	1	1	yes	characteristic
106	C₆₇⋊#95	C₂×C₄	1	1	yes	characteristic
107	C₆₇⋊#95	C₂×C₄	1	1	yes	characteristic
108	C₆₇⋊#95	C₂×C₄	1	1	yes	characteristic
109	C₆₇⋊#95	C₂×C₄	1	1	yes	characteristic
110	C₆₇⋊#95	C₂×C₄	1	1	yes	characteristic
111	C₆₇⋊#95	C₂×C₄	1	1	yes	characteristic
112	C₆₇⋊#95	C₂×C₄	1	1	yes	characteristic
113	C₆₇⋊#95	C₂×C₄	1	1	yes	characteristic
114	C₆₇⋊#95	C₂×C₄	1	1	yes	characteristic
115	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
116	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
117	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
118	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
119	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
120	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
121	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
122	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
123	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
124	C₄⋊C₈×C₆₇	—	2	2	no	—
125	C₄⋊C₈×C₆₇	—	4	1	no	—
126	C₄⋊C₈×C₆₇	—	4	1	no	—
127	#97×C₆₇	—	2	2	no	—
128	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
129	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
130	C₂³×D₆₇	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	C₂²×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
137	C₂²×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
138	C₂×C₄×D₆₇	—	2	1	no	—
139	C₂×C₄×D₆₇	—	2	1	no	—
140	C₂×C₄×D₆₇	—	2	1	no	—
141	C₂×C₄×D₆₇	—	2	1	no	—
142	C₂×C₄×D₆₇	C₂×D₄	1	2	yes	—
143	C₂×C₄×D₆₇	C₂×Q₈	1	2	yes	—
144	#11761611	—	4	4	no	—
145	C₂×C₆₇⋊C₈	—	2	1	no	—
146	C₂×C₆₇⋊C₈	—	2	1	no	—
147	C₂×C₆₇⋊C₈	—	2	1	no	—
148	C₂×C₆₇⋊C₈	—	2	1	no	—
149	C₂×C₆₇⋊C₈	—	2	1	no	—
150	C₂×C₆₇⋊C₈	—	2	1	no	—
151	C₂×C₆₇⋊C₈	—	2	1	no	—
152	C₂×C₆₇⋊C₈	—	2	1	no	—
153	C₂×C₆₇⋊C₈	—	2	1	no	—
154	C₂×C₆₇⋊C₈	—	2	1	no	—
155	C₂×C₆₇⋊C₈	—	2	1	no	—
156	C₂×C₆₇⋊C₈	—	2	1	no	—
157	C₈×D₆₇	—	4	4	no	—
158	C₄⋊Dic₆₇	—	2	2	no	—
159	C₄⋊Dic₆₇	—	4	1	no	—
160	Dic₆₇⋊C₄	C₄⋊C₄	1	4	yes	—
161	Dic₆₇⋊C₄	C₄⋊C₄	1	4	yes	—
162	(C₂×D₆₇)⋊C₄	—	2	2	no	—
163	(C₂×D₆₇)⋊C₄	—	2	2	no	—
164	(C₂×D₆₇)⋊C₄	—	4	1	no	—
165	(C₂×D₆₇)⋊C₄	—	4	1	no	—
166	(C₂×D₆₇)⋊C₄	C₄⋊C₄	1	4	yes	—
167	(C₂×D₆₇)⋊C₄	C₄⋊C₄	1	4	yes	—
168	C₄×Dic₆₇	—	2	2	no	—
169	C₄×Dic₆₇	—	2	1	no	—
170	C₄×Dic₆₇	—	2	1	no	—
171	C₄×Dic₆₇	—	2	1	no	—
172	C₄×Dic₆₇	—	2	1	no	—
173	C₄×Dic₆₇	—	2	2	no	—
174	C₄×Dic₆₇	—	4	1	no	—
175	C₄×Dic₆₇	—	4	1	no	—
176	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
177	C₄×Dic₆₇	C₂×Q₈	1	2	yes	—
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	C₂²×C₄	1	1	yes	characteristic
184	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
185	C₂×C₈×C₆₇	—	2	1	no	—
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	C₄⋊C₄×C₆₇	—	2	2	no	—
198	C₄⋊C₄×C₆₇	—	4	1	no	—
199	C₄²×C₆₇	—	2	1	no	—
200	C₄²×C₆₇	—	2	1	no	—
201	C₄²×C₆₇	—	2	1	no	—
202	C₄²×C₆₇	—	2	1	no	—
203	C₄²×C₆₇	C₂×D₄	1	2	yes	—
204	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
205	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
206	C₂²×D₆₇	—	2	1	no	—
207	C₂²×D₆₇	—	2	1	no	—
208	C₂²×D₆₇	—	2	1	no	—
209	C₂²×D₆₇	—	2	1	no	—
210	C₂²×D₆₇	—	2	1	no	—
211	C₂²×D₆₇	—	2	1	no	—
212	C₂²×D₆₇	C₂×C₄⋊C₄	1	2	yes	—
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	1	no	—
220	C₂×C₄×C₆₇	—	2	1	no	—
221	C₂×C₄×C₆₇	—	2	1	no	—
222	C₂×C₄×C₆₇	—	2	1	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	C₂×C₄⋊C₄	1	2	yes	—
229	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
230	C₂×C₄×C₆₇	#131	1	2	yes	—
231	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
232	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
233	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
234	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
235	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
236	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
237	C₂×Dic₆₇	—	2	1	no	—
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	1	no	—
253	C₂×Dic₆₇	—	2	1	no	—
254	C₂×Dic₆₇	C₂×C₄⋊C₄	1	2	yes	—
255	C₂×Dic₆₇	C₂×C₄⋊C₄	1	2	yes	—
256	C₂×Dic₆₇	C₂×C₄⋊C₄	1	2	yes	—
257	C₂×Dic₆₇	#131	1	2	yes	—
258	C₄×D₆₇	—	2	2	no	—
259	C₄×D₆₇	—	2	2	no	—
260	C₄×D₆₇	—	2	2	no	—
261	C₄×D₆₇	—	2	2	no	—
262	C₈×C₆₇	—	4	1	no	—
263	C₈×C₆₇	—	4	1	no	—
264	C₈×C₆₇	—	4	1	no	—
265	C₈×C₆₇	—	4	1	no	—
266	C₆₇⋊C₈	—	4	1	no	—
267	C₆₇⋊C₈	—	4	1	no	—
268	C₆₇⋊C₈	—	4	1	no	—
269	C₆₇⋊C₈	—	4	1	no	—
270	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
271	C₂²×C₆₇	#520	1	1	yes	characteristic
272	C₂²×C₆₇	#520	1	1	yes	characteristic
273	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
274	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
275	C₂²×C₆₇	#568	1	1	yes	characteristic
276	C₂²×C₆₇	#568	1	1	yes	characteristic
277	C₂×D₆₇	—	2	2	no	—
278	C₂×D₆₇	—	2	2	no	—
279	C₂×D₆₇	—	2	2	no	—
280	C₂×D₆₇	—	4	1	no	—
281	C₂×D₆₇	—	4	1	no	—
282	C₂×D₆₇	—	4	1	no	—
283	C₂×D₆₇	—	4	1	no	—
284	C₄×C₆₇	—	2	2	no	—
285	C₄×C₆₇	—	2	2	no	—
286	C₄×C₆₇	—	2	2	no	—
287	C₄×C₆₇	—	4	1	no	—
288	C₄×C₆₇	—	4	1	no	—
289	C₄×C₆₇	—	4	1	no	—
290	C₄×C₆₇	#535	1	1	yes	characteristic
291	C₄×C₆₇	#536	1	1	yes	characteristic
292	C₄×C₆₇	#544	1	1	yes	characteristic
293	C₄×C₆₇	#544	1	1	yes	characteristic
294	Dic₆₇	—	2	2	no	—
295	Dic₆₇	—	2	2	no	—
296	Dic₆₇	—	2	2	no	—
297	Dic₆₇	—	2	2	no	—
298	Dic₆₇	—	4	1	no	—
299	Dic₆₇	—	4	1	no	—
300	Dic₆₇	—	4	1	no	—
301	#21980	—	67	1	no	maximal, Sylow
302	C₂×C₆₇	#2317	1	1	yes	characteristic
303	C₂×C₆₇	#2906	1	1	yes	characteristic
304	C₂×C₆₇	#2906	1	1	yes	characteristic
305	C₂×C₆₇	#2992	1	1	yes	characteristic
306	C₂×C₆₇	#2993	1	1	yes	characteristic
307	C₂×C₆₇	#2994	1	1	yes	characteristic
308	C₂×C₆₇	#2995	1	1	yes	characteristic
309	D₆₇	—	4	2	no	—
310	C₂×#433	—	67	1	no	—
311	C₂×#424	—	67	1	no	—
312	#2298	—	67	1	no	—
313	#1940	—	67	1	no	—
314	#1940	D₆₇	1	1	yes	characteristic, complemented
315	#1939	—	67	1	no	—
316	#1939	—	67	1	no	—
317	#1851	—	67	1	no	—
318	#1851	—	67	1	no	—
319	#1850	—	67	1	no	—
320	#1850	—	67	1	no	—
321	#1843	—	67	1	no	—
322	#1842	—	67	1	no	—
323	#1832	—	67	1	no	—
324	#1831	—	67	1	no	—
325	C₆₇	#21980	1	1	yes	characteristic, complemented, Sylow
326	C₂²×M₁₆	—	67	1	no	—
327	C₂³×C₈	—	67	1	no	—
328	C₂×#117	—	67	1	no	—
329	#433	—	134	4	no	—
330	#424	—	134	4	no	—
331	C₂×C₄⋊C₈	—	67	1	no	—
332	C₂×C₄⋊C₈	—	67	1	no	—
333	C₂×C₄⋊C₈	—	67	1	no	—
334	C₂×C₄⋊C₈	—	67	1	no	—
335	C₂×C₄⋊C₈	—	67	1	no	—
336	C₂×C₄⋊C₈	—	67	1	no	—
337	C₂×C₄⋊C₈	—	67	1	no	—
338	C₂×C₄⋊C₈	—	67	1	no	—
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	C₂×D₆₇	1	1	yes	characteristic
343	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
344	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
345	C₂×C₂²⋊C₈	—	67	1	no	—
346	C₂×C₂²⋊C₈	—	67	1	no	—
347	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
348	C₂×C₄×C₈	—	67	1	no	—
349	#388	—	67	1	no	—
350	#388	—	67	1	no	—
351	#388	—	67	1	no	—
352	#388	—	67	1	no	—
353	#387	—	67	1	no	—
354	#387	—	67	1	no	—
355	#387	—	67	1	no	—
356	#387	—	67	1	no	—
357	C₄×C₄⋊C₄	—	67	1	no	—
358	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
359	C₄×C₂²⋊C₄	—	67	1	no	—
360	C₄×C₂²⋊C₄	—	67	1	no	—
361	#376	—	67	1	no	—
362	C₄³	—	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	C₂×M₁₆	—	268	1	no	—
373	C₂×M₁₆	—	268	1	no	—
374	C₂×M₁₆	—	134	2	no	—
375	C₂²×C₈	—	268	1	no	—
376	C₂²×C₈	—	268	1	no	—
377	C₂²×C₈	—	67	1	no	—
378	C₂²×C₈	—	67	1	no	—
379	C₂²×C₈	—	67	1	no	—
380	C₂²×C₈	—	67	1	no	—
381	C₂²×C₈	—	134	2	no	—
382	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
383	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
384	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
385	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
386	#117	—	67	4	no	—
387	#117	—	67	4	no	—
388	C₂×C₄⋊C₄	—	67	1	no	—
389	C₂×C₄⋊C₄	—	67	1	no	—
390	C₂×C₄⋊C₄	—	67	1	no	—
391	C₂×C₄⋊C₄	—	67	1	no	—
392	C₂×C₄⋊C₄	C₄×D₆₇	1	1	yes	characteristic
393	C₂×C₄⋊C₄	C₄×D₆₇	1	1	yes	characteristic
394	C₂×C₂²⋊C₄	—	67	1	no	—
395	C₂×C₂²⋊C₄	—	67	1	no	—
396	C₂×C₂²⋊C₄	—	67	1	no	—
397	C₂×C₂²⋊C₄	—	67	1	no	—
398	C₂×C₂²⋊C₄	—	67	1	no	—
399	C₂×C₂²⋊C₄	—	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	C₂²×D₆₇	1	1	yes	characteristic
408	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
409	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
410	C₄⋊C₈	—	2	2	no	—
411	C₄⋊C₈	—	4	1	no	—
412	C₄⋊C₈	—	4	1	no	—
413	C₄⋊C₈	—	268	1	no	—
414	C₄⋊C₈	—	268	1	no	—
415	C₄⋊C₈	—	268	1	no	—
416	C₄⋊C₈	—	268	1	no	—
417	C₄⋊C₈	—	268	1	no	—
418	C₄⋊C₈	—	268	1	no	—
419	C₄⋊C₈	—	134	2	no	—
420	C₄⋊C₈	—	134	2	no	—
421	C₄⋊C₈	—	134	2	no	—
422	C₄⋊C₈	—	134	2	no	—
423	C₄⋊C₈	—	134	2	no	—
424	C₂²⋊C₈	—	134	2	no	—
425	C₂²⋊C₈	—	134	2	no	—
426	#97	—	2	2	no	—
427	C₄×C₈	—	134	2	no	—
428	#95	—	67	1	no	—
429	#95	—	67	1	no	—
430	#95	—	67	1	no	—
431	#95	—	67	1	no	—
432	#95	—	67	1	no	—
433	#95	—	67	1	no	—
434	#95	—	67	1	no	—
435	#95	—	67	1	no	—
436	#95	—	67	1	no	—
437	#95	—	67	1	no	—
438	#95	C₄×D₆₇	1	1	yes	characteristic
439	#95	C₄×D₆₇	1	1	yes	characteristic
440	C₂⁴	—	67	1	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	2	no	—
446	C₂²×C₄	—	67	2	no	—
447	C₂²×C₄	—	67	1	no	—
448	C₂²×C₄	—	67	1	no	—
449	C₂²×C₄	—	67	1	no	—
450	C₂²×C₄	—	134	1	no	—
451	C₂²×C₄	—	134	1	no	—
452	C₂²×C₄	—	134	1	no	—
453	C₂²×C₄	—	134	1	no	—
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	C₂×C₄×D₆₇	1	1	yes	characteristic
459	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
460	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
461	M₁₆	—	268	4	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	—	2	1	no	—
473	C₂×C₈	—	2	1	no	—
474	C₂×C₈	—	268	4	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	—	134	1	no	—
487	C₄⋊C₄	—	2	2	no	—
488	C₄⋊C₄	—	4	1	no	—
489	C₄⋊C₄	—	268	1	no	—
490	C₄⋊C₄	—	67	4	no	—
491	C₄⋊C₄	—	67	4	no	—
492	C₄⋊C₄	—	134	2	no	—
493	C₂²⋊C₄	—	268	1	no	—
494	C₂²⋊C₄	—	268	1	no	—
495	C₂²⋊C₄	—	67	4	no	—
496	C₂²⋊C₄	—	67	4	no	—
497	C₂²⋊C₄	—	134	2	no	—
498	C₂²⋊C₄	—	134	2	no	—
499	C₄²	—	2	1	no	—
500	C₄²	—	2	1	no	—
501	C₄²	—	2	1	no	—
502	C₄²	—	2	1	no	—
503	C₄²	—	268	1	no	—
504	C₄²	—	268	1	no	—
505	C₄²	—	67	2	no	—
506	C₄²	—	67	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	D₄×D₆₇	1	2	yes	—
514	C₄²	Q₈×D₆₇	1	2	yes	—
515	C₂³	—	67	2	no	—
516	C₂³	—	134	1	no	—
517	C₂³	—	134	1	no	—
518	C₂³	—	134	1	no	—
519	C₂³	—	134	1	no	—
520	C₂³	—	134	1	no	—
521	C₂³	—	134	1	no	—
522	C₂³	C₂²×C₄×D₆₇	1	1	yes	characteristic
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	1	no	—
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	1	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	67	2	no	—
539	C₂×C₄	—	67	2	no	—
540	C₂×C₄	—	67	2	no	—
541	C₂×C₄	—	67	2	no	—
542	C₂×C₄	—	134	1	no	—
543	C₂×C₄	—	134	1	no	—
544	C₂×C₄	—	134	1	no	—
545	C₂×C₄	—	134	1	no	—
546	C₂×C₄	—	134	1	no	—
547	C₂×C₄	—	134	1	no	—
548	C₂×C₄	—	134	1	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	2	no	—
560	C₂×C₄	—	134	2	no	—
561	C₂×C₄	—	134	2	no	—
562	C₂×C₄	—	134	2	no	—
563	C₂×C₄	C₄⋊C₄×D₆₇	1	2	yes	—
564	C₂×C₄	C₄⋊C₄×D₆₇	1	2	yes	—
565	C₂×C₄	#14524051	1	2	yes	—
566	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
567	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
568	C₂×C₄	#14524088	1	1	yes	characteristic
569	C₂×C₄	#14524089	1	1	yes	characteristic
570	C₂×C₄	#14524090	1	1	yes	characteristic
571	C₂×C₄	#14524091	1	1	yes	characteristic
572	C₈	—	4	1	no	—
573	C₈	—	4	1	no	—
574	C₈	—	4	1	no	—
575	C₈	—	4	1	no	—
576	C₈	—	268	1	no	—
577	C₈	—	268	1	no	—
578	C₈	—	268	1	no	—
579	C₈	—	268	1	no	—
580	C₂²	—	268	1	no	—
581	C₂²	—	268	1	no	—
582	C₂²	—	268	1	no	—
583	C₂²	—	268	1	no	—
584	C₂²	—	134	2	no	—
585	C₂²	—	134	2	no	—
586	C₂²	—	134	2	no	—
587	C₂²	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
588	C₂²	#16863875	1	1	yes	characteristic
589	C₂²	#16863878	1	1	yes	characteristic
590	C₂²	#131×D₆₇	1	1	yes	characteristic
591	C₂²	C₂×#14524051	1	1	yes	characteristic
592	C₂²	#16864161	1	1	yes	characteristic
593	C₂²	#16864162	1	1	yes	characteristic
594	C₄	—	2	2	no	—
595	C₄	—	2	2	no	—
596	C₄	—	2	2	no	—
597	C₄	—	4	1	no	—
598	C₄	—	4	1	no	—
599	C₄	—	4	1	no	—
600	C₄	—	268	1	no	—
601	C₄	—	268	1	no	—
602	C₄	—	268	1	no	—
603	C₄	—	134	2	no	—
604	C₄	—	134	2	no	—
605	C₄	—	134	2	no	—
606	C₄	—	134	2	no	—
607	C₄	#16863962	1	1	yes	characteristic
608	C₄	#16863969	1	1	yes	characteristic
609	C₄	#16864023	1	1	yes	characteristic
610	C₄	#16864026	1	1	yes	characteristic
611	C₂	—	268	2	no	—
612	C₂	#22562831	1	1	yes	characteristic
613	C₂	#22568776	1	1	yes	characteristic
614	C₂	#22568778	1	1	yes	characteristic
615	C₂	#22569347	1	1	yes	characteristic
616	C₂	#22569361	1	1	yes	characteristic
617	C₂	#22569362	1	1	yes	characteristic
618	C₂	#22569380	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator, socle	C₂³×C₆₇	C₂³×C₄
Fitting	C₆₇×#1940	C₂
prime radical	C₂³×D₆₇	C₂²×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1940	D₆₇
hypocenter	C₆₇	#21980
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ #1940

Complements

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

N	H ≅ G/N	type
4. #22560932	C₂	semidirect
5. #22560928	C₂	semidirect
6. #22560925	C₂	semidirect
11. #22560362	C₂	semidirect
12. #22560360	C₂	semidirect
13. #22560327	C₂	semidirect
14. #22560324	C₂	semidirect
15. C₆₇×#1940	C₂	semidirect
314. #1940	D₆₇	semidirect
325. C₆₇	#21980	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863458	2
C₂×#16863421	2
#22562702	2
#22560932	2
#22560928	2
#22560925	2
#22560393	2
#22560390	2
#22560388	2
#22560385	2
#22560362	2
#22560360	2
#22560327	2
#22560324	2
C₆₇×#1940	2
#21980	67