Subgroups — #30873217

Order 17152.

864 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560768	C₂	1	2	yes	maximal, complemented
2	#22560768	C₂	1	2	yes	maximal, complemented
3	#22560389	C₂	1	2	yes	maximal
4	#22560352	C₂	1	1	yes	maximal, characteristic, complemented
5	#22560326	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560277	C₂	1	1	yes	maximal, characteristic
7	C₄×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²⋊C₄×C₆₇⋊C₈	C₂	1	2	yes	maximal
9	#22559955	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#16863151	C₂	1	1	yes	maximal, characteristic
11	C₄×#117×C₆₇	C₂	1	1	yes	maximal, characteristic
12	C₂³×C₆₇⋊C₈	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
15	C₂×C₂²⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
16	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
17	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
18	#16863151	C₂²	1	2	yes	—
19	#16863151	C₂²	1	2	yes	—
20	#16863151	C₂²	1	2	yes	—
21	#16863151	C₂²	1	2	yes	—
22	#16863151	C₂²	1	2	yes	—
23	#16863151	C₂²	1	2	yes	—
24	#16863151	C₂²	1	1	yes	characteristic
25	#16863151	C₂²	1	1	yes	characteristic
26	#16863151	C₂²	1	1	yes	characteristic
27	#16863151	C₂²	1	1	yes	characteristic
28	C₂×#117×C₆₇	C₄	1	1	yes	characteristic
29	C₂×#117×C₆₇	C₄	1	1	yes	characteristic
30	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
31	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
33	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
35	C₄×C₂²⋊C₄×C₆₇	C₄	1	1	yes	characteristic
36	C₄×C₂²⋊C₄×C₆₇	C₄	1	1	yes	characteristic
37	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
38	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
39	#376×C₆₇	C₄	1	1	yes	characteristic
40	#376×C₆₇	C₄	1	1	yes	characteristic
41	C₄³×C₆₇	C₂²	1	1	yes	characteristic
42	C₄³×C₆₇	C₂²	1	1	yes	characteristic
43	C₂²×C₆₇⋊C₈	—	2	4	no	—
44	C₂²×C₆₇⋊C₈	—	2	4	no	—
45	C₂²×C₆₇⋊C₈	—	4	2	no	—
46	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
47	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
48	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
49	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
50	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
51	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
52	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
53	C₄×C₆₇⋊C₈	—	2	4	no	—
54	C₄×C₆₇⋊C₈	—	4	2	no	—
55	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
57	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
58	#117×C₆₇	—	2	4	no	—
59	#117×C₆₇	C₈	1	8	yes	complemented
60	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
65	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
66	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
67	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₆₇	—	2	4	no	—
69	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
78	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
79	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
80	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
81	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
82	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
83	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
84	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
85	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
86	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
87	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
88	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
89	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
90	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
91	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
92	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
93	C₂×C₆₇⋊C₈	—	2	2	no	—
94	C₂×C₆₇⋊C₈	—	2	2	no	—
95	C₂×C₆₇⋊C₈	—	2	2	no	—
96	C₂×C₆₇⋊C₈	—	2	2	no	—
97	C₂×C₆₇⋊C₈	—	2	2	no	—
98	C₂×C₆₇⋊C₈	—	2	2	no	—
99	C₂×C₆₇⋊C₈	—	2	2	no	—
100	C₂×C₆₇⋊C₈	—	2	2	no	—
101	C₂×C₆₇⋊C₈	—	2	2	no	—
102	C₂×C₆₇⋊C₈	—	2	2	no	—
103	C₂×C₆₇⋊C₈	—	2	2	no	—
104	C₂×C₆₇⋊C₈	—	4	8	no	—
105	C₂×C₆₇⋊C₈	#41	1	4	yes	—
106	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₄×C₆₇	—	2	2	no	—
108	C₂²×C₄×C₆₇	—	2	2	no	—
109	C₂²×C₄×C₆₇	—	2	2	no	—
110	C₂²×C₄×C₆₇	—	2	1	no	—
111	C₂²×C₄×C₆₇	—	2	1	no	—
112	C₂²×C₄×C₆₇	—	2	2	no	—
113	C₂²×C₄×C₆₇	—	2	2	no	—
114	C₂²×C₄×C₆₇	—	4	1	no	—
115	C₂²×C₄×C₆₇	C₂×C₈	1	4	yes	—
116	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
117	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
120	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
121	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
122	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
123	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
124	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
125	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
126	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
127	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
128	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
129	C₂²×C₄×C₆₇	#41	1	4	yes	—
130	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
131	C₄⋊C₄×C₆₇	—	2	2	no	—
132	C₄⋊C₄×C₆₇	—	2	2	no	—
133	C₄⋊C₄×C₆₇	C₂×C₈	1	4	yes	complemented
134	C₄⋊C₄×C₆₇	C₂×C₈	1	4	yes	complemented
135	C₂²⋊C₄×C₆₇	—	2	2	no	—
136	C₂²⋊C₄×C₆₇	C₂×C₈	1	4	yes	—
137	C₂²⋊C₄×C₆₇	C₂×C₈	1	4	yes	—
138	C₂²⋊C₄×C₆₇	M₁₆	1	4	yes	—
139	C₄²×C₆₇	—	2	2	no	—
140	C₄²×C₆₇	—	2	2	no	—
141	C₄²×C₆₇	—	2	2	no	—
142	C₄²×C₆₇	—	2	2	no	—
143	C₄²×C₆₇	—	2	2	no	—
144	C₄²×C₆₇	—	2	2	no	—
145	C₄²×C₆₇	—	2	2	no	—
146	C₄²×C₆₇	—	2	2	no	—
147	C₄²×C₆₇	—	2	2	no	—
148	C₄²×C₆₇	—	2	2	no	—
149	C₄²×C₆₇	—	2	1	no	—
150	C₄²×C₆₇	—	2	1	no	—
151	C₄²×C₆₇	—	2	1	no	—
152	C₄²×C₆₇	—	2	1	no	—
153	C₄²×C₆₇	—	4	1	no	—
154	C₄²×C₆₇	—	4	1	no	—
155	C₄²×C₆₇	C₂×C₈	1	4	yes	complemented
156	C₄²×C₆₇	C₂×C₈	1	4	yes	complemented
157	C₂³×C₆₇	—	2	1	no	—
158	C₂³×C₆₇	—	2	1	no	—
159	C₂³×C₆₇	—	2	1	no	—
160	C₂³×C₆₇	—	2	1	no	—
161	C₂³×C₆₇	#117	1	2	yes	—
162	C₂³×C₆₇	C₂²×C₈	1	2	yes	—
163	C₂³×C₆₇	C₂×M₁₆	1	2	yes	—
164	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
165	C₂×C₄×C₆₇	—	2	1	no	—
166	C₂×C₄×C₆₇	—	2	1	no	—
167	C₂×C₄×C₆₇	—	2	1	no	—
168	C₂×C₄×C₆₇	—	2	1	no	—
169	C₂×C₄×C₆₇	—	2	1	no	—
170	C₂×C₄×C₆₇	—	2	1	no	—
171	C₂×C₄×C₆₇	—	2	1	no	—
172	C₂×C₄×C₆₇	—	2	1	no	—
173	C₂×C₄×C₆₇	—	2	1	no	—
174	C₂×C₄×C₆₇	—	2	1	no	—
175	C₂×C₄×C₆₇	—	2	1	no	—
176	C₂×C₄×C₆₇	—	2	1	no	—
177	C₂×C₄×C₆₇	—	2	1	no	—
178	C₂×C₄×C₆₇	—	2	1	no	—
179	C₂×C₄×C₆₇	—	2	1	no	—
180	C₂×C₄×C₆₇	—	2	1	no	—
181	C₂×C₄×C₆₇	—	2	1	no	—
182	C₂×C₄×C₆₇	—	2	1	no	—
183	C₂×C₄×C₆₇	—	2	1	no	—
184	C₂×C₄×C₆₇	—	2	1	no	—
185	C₂×C₄×C₆₇	—	2	8	no	—
186	C₂×C₄×C₆₇	—	2	4	no	—
187	C₂×C₄×C₆₇	—	2	4	no	—
188	C₂×C₄×C₆₇	—	2	1	no	—
189	C₂×C₄×C₆₇	—	2	1	no	—
190	C₂×C₄×C₆₇	—	2	1	no	—
191	C₂×C₄×C₆₇	—	2	1	no	—
192	C₂×C₄×C₆₇	—	2	1	no	—
193	C₂×C₄×C₆₇	—	2	1	no	—
194	C₂×C₄×C₆₇	—	2	1	no	—
195	C₂×C₄×C₆₇	—	2	1	no	—
196	C₂×C₄×C₆₇	—	2	1	no	—
197	C₂×C₄×C₆₇	—	4	4	no	—
198	C₂×C₄×C₆₇	#117	1	2	yes	—
199	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
200	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
201	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
202	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
203	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
204	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
205	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
206	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
207	C₂×C₄×C₆₇	#131	1	2	yes	—
208	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
209	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
210	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
211	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
212	C₆₇⋊C₈	—	2	4	no	—
213	C₆₇⋊C₈	—	4	2	no	—
214	C₆₇⋊C₈	—	4	2	no	—
215	C₆₇⋊C₈	—	4	2	no	—
216	C₂²×C₆₇	—	2	2	no	—
217	C₂²×C₆₇	—	2	2	no	—
218	C₂²×C₆₇	—	2	2	no	—
219	C₂²×C₆₇	—	2	2	no	—
220	C₂²×C₆₇	—	2	2	no	—
221	C₂²×C₆₇	—	2	2	no	—
222	C₂²×C₆₇	#431	1	4	yes	—
223	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
224	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
225	C₂²×C₆₇	#520	1	1	yes	characteristic
226	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
227	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
228	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
229	C₂²×C₆₇	#568	1	1	yes	characteristic
230	C₄×C₆₇	—	2	2	no	—
231	C₄×C₆₇	—	2	2	no	—
232	C₄×C₆₇	—	2	2	no	—
233	C₄×C₆₇	—	2	2	no	—
234	C₄×C₆₇	—	2	2	no	—
235	C₄×C₆₇	—	2	2	no	—
236	C₄×C₆₇	—	2	2	no	—
237	C₄×C₆₇	—	2	2	no	—
238	C₄×C₆₇	—	2	2	no	—
239	C₄×C₆₇	—	2	2	no	—
240	C₄×C₆₇	—	4	1	no	—
241	C₄×C₆₇	—	4	1	no	—
242	C₄×C₆₇	—	4	1	no	—
243	C₄×C₆₇	#551	1	2	yes	—
244	C₄×C₆₇	#552	1	2	yes	—
245	#22589	—	67	1	no	maximal, Sylow
246	C₂×C₆₇	—	2	4	no	—
247	C₂×C₆₇	#2343	1	1	yes	characteristic
248	C₂×C₆₇	C₂×#431	1	1	yes	characteristic
249	C₂×C₆₇	#2924	1	1	yes	characteristic
250	C₂×C₆₇	C₈×#41	1	1	yes	characteristic
251	C₂×C₆₇	#2970	1	1	yes	characteristic
252	C₂×C₆₇	#2988	1	1	yes	characteristic
253	C₂×C₆₇	#2990	1	1	yes	characteristic
254	C₄×#117	D₆₇	1	1	yes	characteristic
255	#1916	—	67	2	no	—
256	#1916	—	67	2	no	—
257	#1850	—	67	2	no	—
258	#1839	—	67	1	no	—
259	#1831	—	67	1	no	—
260	#1817	—	67	1	no	—
261	C₄×C₄⋊C₈	—	67	1	no	—
262	C₈×C₂²⋊C₄	—	67	2	no	—
263	#1744	—	67	1	no	—
264	C₂×#336	—	67	1	no	—
265	C₆₇	#22589	1	1	yes	characteristic, complemented, Sylow
266	C₂³×C₈	—	67	2	no	—
267	C₂×#117	Dic₆₇	1	1	yes	characteristic
268	C₂×#117	Dic₆₇	1	1	yes	characteristic
269	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
270	C₂×C₄⋊C₈	—	67	2	no	—
271	C₂×C₄⋊C₈	—	67	2	no	—
272	C₂×C₂²⋊C₈	—	67	2	no	—
273	C₂×C₄×C₈	—	67	2	no	—
274	C₂×C₄×C₈	—	67	2	no	—
275	C₄×C₄⋊C₄	Dic₆₇	1	1	yes	characteristic
276	C₄×C₄⋊C₄	Dic₆₇	1	1	yes	characteristic
277	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
278	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
279	C₄×C₂²⋊C₄	Dic₆₇	1	1	yes	characteristic
280	C₄×C₂²⋊C₄	Dic₆₇	1	1	yes	characteristic
281	C₄×C₂²⋊C₄	C₂×D₆₇	1	1	yes	characteristic
282	C₄×C₂²⋊C₄	C₂×D₆₇	1	1	yes	characteristic
283	#376	Dic₆₇	1	1	yes	characteristic
284	#376	Dic₆₇	1	1	yes	characteristic
285	C₄³	C₂×D₆₇	1	1	yes	characteristic
286	C₄³	C₂×D₆₇	1	1	yes	characteristic
287	#336	—	67	1	no	—
288	#336	—	67	1	no	—
289	#336	—	67	2	no	—
290	#336	—	67	1	no	—
291	#336	—	67	2	no	—
292	#336	—	67	2	no	—
293	#336	—	67	2	no	—
294	#336	—	67	2	no	—
295	#336	—	67	2	no	—
296	#336	—	67	1	no	—
297	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
298	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
299	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
300	C₂²×C₈	—	268	2	no	—
301	C₂²×C₈	—	67	2	no	—
302	C₂²×C₈	—	67	2	no	—
303	C₂²×C₈	—	67	2	no	—
304	C₂²×C₈	—	67	2	no	—
305	C₂²×C₈	—	134	4	no	—
306	C₂²×C₈	—	134	4	no	—
307	#117	—	2	4	no	—
308	#117	C₆₇⋊C₈	1	8	yes	complemented
309	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
310	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
311	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
312	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
313	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
314	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
315	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
316	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
317	C₂×C₄²	—	2	4	no	—
318	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
319	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
320	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
321	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
322	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
323	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
324	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
325	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
326	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
327	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
328	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
329	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
330	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
331	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
332	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
333	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
334	C₄⋊C₈	—	268	2	no	—
335	C₄⋊C₈	—	134	4	no	—
336	C₂²⋊C₈	—	134	4	no	—
337	C₄×C₈	—	268	2	no	—
338	C₄×C₈	—	134	4	no	—
339	#95	C₂×Dic₆₇	1	1	yes	characteristic
340	#95	C₂×Dic₆₇	1	1	yes	characteristic
341	#95	C₂×Dic₆₇	1	1	yes	characteristic
342	#95	C₂×Dic₆₇	1	1	yes	characteristic
343	#95	C₂×Dic₆₇	1	1	yes	characteristic
344	#95	C₂×Dic₆₇	1	1	yes	characteristic
345	#95	C₂×Dic₆₇	1	1	yes	characteristic
346	#95	C₂×Dic₆₇	1	1	yes	characteristic
347	C₂⁴	C₂²×Dic₆₇	1	1	yes	characteristic
348	C₂²×C₄	—	2	2	no	—
349	C₂²×C₄	—	2	2	no	—
350	C₂²×C₄	—	2	2	no	—
351	C₂²×C₄	—	2	1	no	—
352	C₂²×C₄	—	2	1	no	—
353	C₂²×C₄	—	2	2	no	—
354	C₂²×C₄	—	2	2	no	—
355	C₂²×C₄	—	4	1	no	—
356	C₂²×C₄	C₂×C₆₇⋊C₈	1	4	yes	—
357	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
358	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
359	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
360	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
361	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
362	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
363	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
364	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
365	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
366	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
367	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
368	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
369	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
370	C₂²×C₄	#11761628	1	4	yes	—
371	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
372	C₂×C₈	—	268	8	no	—
373	C₂×C₈	—	67	4	no	—
374	C₂×C₈	—	134	2	no	—
375	C₂×C₈	—	134	2	no	—
376	C₂×C₈	—	134	2	no	—
377	C₂×C₈	—	134	2	no	—
378	C₂×C₈	—	134	2	no	—
379	C₂×C₈	—	134	2	no	—
380	C₂×C₈	—	134	2	no	—
381	C₂×C₈	—	134	2	no	—
382	C₂×C₈	—	134	2	no	—
383	C₂×C₈	—	134	2	no	—
384	C₂×C₈	—	134	2	no	—
385	C₄⋊C₄	—	2	2	no	—
386	C₄⋊C₄	—	2	2	no	—
387	C₄⋊C₄	C₂×C₆₇⋊C₈	1	4	yes	complemented
388	C₄⋊C₄	C₂×C₆₇⋊C₈	1	4	yes	complemented
389	C₂²⋊C₄	—	2	2	no	—
390	C₂²⋊C₄	C₂×C₆₇⋊C₈	1	4	yes	—
391	C₂²⋊C₄	C₂×C₆₇⋊C₈	1	4	yes	—
392	C₂²⋊C₄	#11761612	1	4	yes	—
393	C₄²	—	2	2	no	—
394	C₄²	—	2	2	no	—
395	C₄²	—	2	2	no	—
396	C₄²	—	2	2	no	—
397	C₄²	—	2	2	no	—
398	C₄²	—	2	2	no	—
399	C₄²	—	2	2	no	—
400	C₄²	—	2	2	no	—
401	C₄²	—	2	2	no	—
402	C₄²	—	2	2	no	—
403	C₄²	—	2	1	no	—
404	C₄²	—	2	1	no	—
405	C₄²	—	2	1	no	—
406	C₄²	—	2	1	no	—
407	C₄²	—	4	1	no	—
408	C₄²	—	4	1	no	—
409	C₄²	C₂×C₆₇⋊C₈	1	4	yes	complemented
410	C₄²	C₂×C₆₇⋊C₈	1	4	yes	complemented
411	C₂³	—	2	1	no	—
412	C₂³	—	2	1	no	—
413	C₂³	—	2	1	no	—
414	C₂³	—	2	1	no	—
415	C₂³	#14524002	1	2	yes	—
416	C₂³	C₂²×C₆₇⋊C₈	1	2	yes	—
417	C₂³	C₂×#11761612	1	2	yes	—
418	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	1	no	—
423	C₂×C₄	—	2	1	no	—
424	C₂×C₄	—	2	1	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	2	1	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	2	1	no	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	1	no	—
435	C₂×C₄	—	2	1	no	—
436	C₂×C₄	—	2	1	no	—
437	C₂×C₄	—	2	1	no	—
438	C₂×C₄	—	2	1	no	—
439	C₂×C₄	—	2	8	no	—
440	C₂×C₄	—	2	4	no	—
441	C₂×C₄	—	2	4	no	—
442	C₂×C₄	—	2	1	no	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	2	1	no	—
445	C₂×C₄	—	2	1	no	—
446	C₂×C₄	—	2	1	no	—
447	C₂×C₄	—	2	1	no	—
448	C₂×C₄	—	2	1	no	—
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	4	4	no	—
452	C₂×C₄	#14524002	1	2	yes	—
453	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
454	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
455	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
456	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
457	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
458	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
459	C₂×C₄	C₂×#11761612	1	2	yes	—
460	C₂×C₄	C₂×#11761612	1	2	yes	—
461	C₂×C₄	#14524050	1	2	yes	—
462	C₂×C₄	#41×D₆₇	1	2	yes	—
463	C₂×C₄	C₂×#11761628	1	2	yes	—
464	C₂×C₄	#14524089	1	1	yes	characteristic
465	C₂×C₄	#14524090	1	1	yes	characteristic
466	C₈	—	268	2	no	—
467	C₈	—	268	2	no	—
468	C₈	—	268	2	no	—
469	C₈	—	134	4	no	—
470	C₂²	—	2	2	no	—
471	C₂²	—	2	2	no	—
472	C₂²	—	2	2	no	—
473	C₂²	—	2	2	no	—
474	C₂²	—	2	2	no	—
475	C₂²	—	2	2	no	—
476	C₂²	#16863449	1	4	yes	—
477	C₂²	C₂×#14524002	1	1	yes	characteristic
478	C₂²	#41×Dic₆₇	1	1	yes	characteristic
479	C₂²	#16863879	1	1	yes	characteristic
480	C₂²	C₂³×C₆₇⋊C₈	1	1	yes	characteristic
481	C₂²	C₂²×#11761612	1	1	yes	characteristic
482	C₂²	C₂×#14524050	1	1	yes	characteristic
483	C₂²	#16864163	1	1	yes	characteristic
484	C₄	—	2	2	no	—
485	C₄	—	2	2	no	—
486	C₄	—	2	2	no	—
487	C₄	—	2	2	no	—
488	C₄	—	2	2	no	—
489	C₄	—	2	2	no	—
490	C₄	—	2	2	no	—
491	C₄	—	2	2	no	—
492	C₄	—	2	2	no	—
493	C₄	—	2	2	no	—
494	C₄	—	4	1	no	—
495	C₄	—	4	1	no	—
496	C₄	—	4	1	no	—
497	C₄	#16864070	1	2	yes	—
498	C₄	#16864079	1	2	yes	—
499	C₂	—	2	4	no	—
500	C₂	#22563069	1	1	yes	characteristic
501	C₂	C₂×#16863449	1	1	yes	characteristic
502	C₂	#22568867	1	1	yes	characteristic
503	C₂	#41×C₆₇⋊C₈	1	1	yes	characteristic
504	C₂	#22569223	1	1	yes	characteristic
505	C₂	#22569320	1	1	yes	characteristic
506	C₂	#22569334	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator	C₂²×C₆₇	C₂³×C₈
Fitting	C₄×#117×C₆₇	C₂
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂⁴×C₆₇	C₂²×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₄×#117	D₆₇
hypocenter	C₆₇	#22589
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
1. #22560768	C₂	semidirect
2. #22560768	C₂	semidirect
4. #22560352	C₂	semidirect
5. #22560326	C₂	semidirect
7. C₄×C₄⋊(C₆₇⋊C₈)	C₂	semidirect
9. #22559955	C₂	semidirect
59. #117×C₆₇	C₈	semidirect
133. C₄⋊C₄×C₆₇	C₂×C₈	semidirect
134. C₄⋊C₄×C₆₇	C₂×C₈	semidirect
155. C₄²×C₆₇	C₂×C₈	semidirect
156. C₄²×C₆₇	C₂×C₈	semidirect
265. C₆₇	#22589	semidirect
308. #117	C₆₇⋊C₈	semidirect
387. C₄⋊C₄	C₂×C₆₇⋊C₈	semidirect
388. C₄⋊C₄	C₂×C₆₇⋊C₈	semidirect
409. C₄²	C₂×C₆₇⋊C₈	semidirect
410. C₄²	C₂×C₆₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#22560768	2
#22560389	2
#22560352	2
#22560326	2
#22560277	2
C₄×C₄⋊(C₆₇⋊C₈)	2
C₂²⋊C₄×C₆₇⋊C₈	2
#22559955	2
C₂×#16863151	2
C₄×#117×C₆₇	2
#22589	67