Subgroups — #30866464

Order 17152.

658 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568844	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863454	C₂	1	1	yes	maximal, characteristic
3	#22568772	C₂	1	1	yes	maximal, characteristic
4	#22560983	C₂	1	1	yes	maximal, characteristic, complemented
5	#22560983	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560981	C₂	1	1	yes	maximal, characteristic, complemented
7	#22560981	C₂	1	1	yes	maximal, characteristic, complemented
8	#22560416	C₂	1	1	yes	maximal, characteristic
9	#22560414	C₂	1	1	yes	maximal, characteristic
10	#22560413	C₂	1	1	yes	maximal, characteristic
11	#22560382	C₂	1	1	yes	maximal, characteristic, complemented
12	#22560379	C₂	1	1	yes	maximal, characteristic, complemented
13	#22560348	C₂	1	1	yes	maximal, characteristic, complemented
14	#22560345	C₂	1	1	yes	maximal, characteristic, complemented
15	C₆₇×#1855	C₂	1	1	yes	maximal, characteristic
16	C₂²×#11761612	C₂²	1	1	yes	characteristic
17	C₂×#14524002	C₂²	1	1	yes	characteristic
18	#16863454	—	2	1	no	—
19	#16863454	—	2	1	no	—
20	#16863454	—	2	1	no	—
21	#16863454	—	2	1	no	—
22	#16863454	C₄	1	2	yes	—
23	#16863454	C₄	1	2	yes	—
24	#16863454	C₄	1	2	yes	—
25	#16863454	C₄	1	2	yes	—
26	C₂×#14523979	C₂²	1	1	yes	characteristic
27	C₂×#14523979	C₂²	1	1	yes	characteristic
28	C₂×#14523978	C₂²	1	1	yes	characteristic
29	C₂×#14523978	C₂²	1	1	yes	characteristic
30	C₂×#14523978	C₂²	1	1	yes	characteristic
31	C₂×#14523978	C₂²	1	1	yes	characteristic
32	#16863424	—	2	1	no	—
33	#16863424	—	2	1	no	—
34	#16863422	—	2	1	no	—
35	#16863422	—	2	1	no	—
36	#16863419	C₂²	1	1	yes	characteristic
37	#16863419	C₂²	1	1	yes	characteristic
38	#16863419	C₂²	1	1	yes	characteristic
39	#16863419	C₂²	1	1	yes	characteristic
40	#16863419	C₂²	1	1	yes	characteristic
41	#16863419	C₂²	1	1	yes	characteristic
42	C₂×Dic₆₇⋊C₈	C₂²	1	1	yes	characteristic
43	C₂×Dic₆₇⋊C₈	C₂²	1	1	yes	characteristic
44	#16863360	C₂²	1	1	yes	characteristic
45	C₂×C₂²⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
46	M₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
47	M₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
48	C₂×#14523953	C₂²	1	1	yes	characteristic
49	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
50	#16863161	C₂²	1	1	yes	characteristic
51	#16863161	C₂²	1	1	yes	characteristic
52	#16863160	C₂²	1	1	yes	characteristic
53	#16863160	C₂²	1	1	yes	characteristic
54	#16863160	C₂²	1	1	yes	characteristic
55	#16863160	C₂²	1	1	yes	characteristic
56	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
57	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
58	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
59	#407×C₆₇	C₂²	1	1	yes	characteristic
60	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
61	#341×C₆₇	C₂²	1	1	yes	characteristic
62	#341×C₆₇	C₂²	1	1	yes	characteristic
63	C₂×#11761612	—	2	1	no	—
64	C₂×#11761612	—	4	1	no	—
65	C₂×#11761612	—	4	1	no	—
66	C₂×#11761612	C₂³	1	1	yes	characteristic
67	C₂×#11761612	C₂³	1	1	yes	characteristic
68	C₂×#11761612	C₂³	1	1	yes	characteristic
69	C₂×#11761612	C₂³	1	1	yes	characteristic
70	C₂²×C₆₇⋊C₈	—	2	1	no	—
71	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
72	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
73	#14524002	C₂×C₄	1	1	yes	characteristic
74	#14524002	C₂×C₄	1	1	yes	characteristic
75	#14524002	C₂×C₄	1	1	yes	characteristic
76	#14524002	C₂×C₄	1	1	yes	characteristic
77	#14524002	D₄	1	1	yes	characteristic
78	#14524002	D₄	1	1	yes	characteristic
79	#14524002	D₄	1	1	yes	characteristic
80	#14524002	D₄	1	1	yes	characteristic
81	C₂×C₄⋊Dic₆₇	Q₈	1	1	yes	characteristic
82	C₂×C₄⋊Dic₆₇	Q₈	1	1	yes	characteristic
83	C₂×C₂²⋊Dic₆₇	Q₈	1	1	yes	characteristic
84	C₂×C₂²⋊Dic₆₇	Q₈	1	1	yes	characteristic
85	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
86	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
87	#14523979	—	4	1	no	—
88	#14523979	—	4	1	no	—
89	#14523978	—	4	1	no	—
90	#14523978	—	4	1	no	—
91	#14523978	—	4	1	no	—
92	#14523978	—	4	1	no	—
93	Dic₆₇⋊C₈	—	2	1	no	—
94	Dic₆₇⋊C₈	—	2	1	no	—
95	Dic₆₇⋊C₈	—	2	1	no	—
96	Dic₆₇⋊C₈	—	2	1	no	—
97	Dic₆₇⋊C₈	—	2	1	no	—
98	Dic₆₇⋊C₈	—	2	1	no	—
99	Dic₆₇⋊C₈	—	2	1	no	—
100	Dic₆₇⋊C₈	—	2	1	no	—
101	Dic₆₇⋊C₈	—	2	1	no	—
102	Dic₆₇⋊C₈	—	2	1	no	—
103	Dic₆₇⋊C₈	—	2	1	no	—
104	Dic₆₇⋊C₈	—	2	1	no	—
105	Dic₆₇⋊C₈	C₂×C₄	1	2	yes	—
106	Dic₆₇⋊C₈	C₂×C₄	1	2	yes	—
107	Dic₆₇⋊C₈	C₂×C₄	1	2	yes	—
108	Dic₆₇⋊C₈	C₂×C₄	1	2	yes	—
109	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
110	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
111	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
112	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
113	#14523953	—	2	1	no	—
114	#14523953	—	2	1	no	—
115	#14523953	C₂×C₄	1	2	yes	—
116	#14523953	C₂×C₄	1	2	yes	—
117	C₈×Dic₆₇	—	2	1	no	—
118	C₈×Dic₆₇	—	2	1	no	—
119	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
120	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
121	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
122	C₂×M₁₆×C₆₇	—	2	1	no	—
123	C₂×M₁₆×C₆₇	—	4	1	no	—
124	C₂×M₁₆×C₆₇	—	4	1	no	—
125	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
126	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
127	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
128	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
129	C₂²×C₈×C₆₇	—	2	1	no	—
130	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
131	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
132	#108×C₆₇	—	4	1	no	—
133	#108×C₆₇	—	4	1	no	—
134	C₂²⋊C₈×C₆₇	—	2	1	no	—
135	C₂²⋊C₈×C₆₇	—	2	1	no	—
136	C₂²⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
137	C₂²⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
138	C₂²×Dic₆₇	C₂×Q₈	1	1	yes	characteristic
139	C₂²×Dic₆₇	C₂×Q₈	1	1	yes	characteristic
140	C₂²×Dic₆₇	C₂×Q₈	1	1	yes	characteristic
141	C₂²×Dic₆₇	C₂×Q₈	1	1	yes	characteristic
142	#11761612	—	2	1	no	—
143	#11761612	—	2	1	no	—
144	#11761612	—	2	1	no	—
145	#11761612	—	2	1	no	—
146	#11761612	—	2	1	no	—
147	#11761612	—	2	1	no	—
148	#11761612	—	2	1	no	—
149	#11761612	—	2	1	no	—
150	#11761612	—	4	2	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	C₂×C₆₇⋊C₈	—	2	1	no	—
158	C₂×C₆₇⋊C₈	—	2	1	no	—
159	C₂×C₆₇⋊C₈	—	4	2	no	—
160	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
161	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
162	C₄⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
163	C₄⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
164	C₄⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
165	C₄⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
166	C₄⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
167	C₄⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
168	C₄⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
169	C₄⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
170	C₂²⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
171	C₂²⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
172	C₂²⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
173	C₂²⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
174	C₂²⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
175	C₂²⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
176	C₂²⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
177	C₂²⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
178	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
179	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
180	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
181	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
182	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
183	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
184	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
185	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
186	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
187	C₂²×C₄×C₆₇	—	2	1	no	—
188	C₂²×C₄×C₆₇	—	2	1	no	—
189	C₂²×C₄×C₆₇	—	2	1	no	—
190	C₂²×C₄×C₆₇	—	2	1	no	—
191	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
192	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
193	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
194	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
195	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
196	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
197	M₁₆×C₆₇	—	2	1	no	—
198	M₁₆×C₆₇	—	2	1	no	—
199	M₁₆×C₆₇	—	2	1	no	—
200	M₁₆×C₆₇	—	2	1	no	—
201	M₁₆×C₆₇	—	2	1	no	—
202	M₁₆×C₆₇	—	2	1	no	—
203	M₁₆×C₆₇	—	2	1	no	—
204	M₁₆×C₆₇	—	2	1	no	—
205	M₁₆×C₆₇	—	4	2	no	—
206	C₂×C₈×C₆₇	—	2	1	no	—
207	C₂×C₈×C₆₇	—	2	1	no	—
208	C₂×C₈×C₆₇	—	2	1	no	—
209	C₂×C₈×C₆₇	—	2	1	no	—
210	C₂×C₈×C₆₇	—	2	1	no	—
211	C₂×C₈×C₆₇	—	2	1	no	—
212	C₂×C₈×C₆₇	—	2	1	no	—
213	C₂×C₈×C₆₇	—	2	1	no	—
214	C₂×C₈×C₆₇	—	4	2	no	—
215	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
216	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
217	C₂³×C₆₇	—	2	1	no	—
218	C₂³×C₆₇	—	2	1	no	—
219	C₂³×C₆₇	—	4	1	no	—
220	C₂³×C₆₇	—	4	1	no	—
221	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
222	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
223	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
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	—	2	1	no	—
230	C₂×C₄×C₆₇	—	2	1	no	—
231	C₂×C₄×C₆₇	—	2	1	no	—
232	C₂×C₄×C₆₇	—	2	1	no	—
233	C₂×C₄×C₆₇	—	2	1	no	—
234	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
235	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
236	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
237	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
238	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
239	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
240	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
241	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
242	C₂×Dic₆₇	—	4	1	no	—
243	C₂×Dic₆₇	—	4	1	no	—
244	C₂×Dic₆₇	—	4	1	no	—
245	C₂×Dic₆₇	—	4	1	no	—
246	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
247	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
248	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
249	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
250	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
251	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
252	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
253	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
254	C₈×C₆₇	—	2	2	no	—
255	C₈×C₆₇	—	2	2	no	—
256	C₈×C₆₇	—	4	1	no	—
257	C₈×C₆₇	—	4	1	no	—
258	C₈×C₆₇	—	4	1	no	—
259	C₈×C₆₇	—	4	1	no	—
260	C₆₇⋊C₈	—	2	2	no	—
261	C₆₇⋊C₈	—	2	2	no	—
262	C₆₇⋊C₈	—	4	1	no	—
263	C₆₇⋊C₈	—	4	1	no	—
264	C₆₇⋊C₈	—	4	1	no	—
265	C₆₇⋊C₈	—	4	1	no	—
266	C₂²×C₆₇	—	2	1	no	—
267	C₂²×C₆₇	—	2	1	no	—
268	C₂²×C₆₇	—	2	1	no	—
269	C₂²×C₆₇	—	2	1	no	—
270	C₂²×C₆₇	—	2	1	no	—
271	C₂²×C₆₇	—	2	1	no	—
272	C₂²×C₆₇	—	2	1	no	—
273	C₂²×C₆₇	—	2	1	no	—
274	C₂²×C₆₇	—	8	1	no	—
275	C₂²×C₆₇	—	8	1	no	—
276	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
277	C₂²×C₆₇	#544	1	1	yes	characteristic
278	C₂²×C₆₇	#544	1	1	yes	characteristic
279	C₄×C₆₇	—	2	1	no	—
280	C₄×C₆₇	—	2	1	no	—
281	C₄×C₆₇	#520	1	1	yes	characteristic
282	C₄×C₆₇	#520	1	1	yes	characteristic
283	C₄×C₆₇	#535	1	1	yes	characteristic
284	C₄×C₆₇	#536	1	1	yes	characteristic
285	Dic₆₇	—	2	1	no	—
286	Dic₆₇	—	2	1	no	—
287	Dic₆₇	—	2	1	no	—
288	Dic₆₇	—	2	1	no	—
289	Dic₆₇	—	2	1	no	—
290	Dic₆₇	—	2	1	no	—
291	Dic₆₇	—	2	1	no	—
292	Dic₆₇	—	2	1	no	—
293	#21995	—	67	1	no	maximal, Sylow
294	C₂×C₆₇	—	2	1	no	—
295	C₂×C₆₇	—	2	1	no	—
296	C₂×C₆₇	—	4	1	no	—
297	C₂×C₆₇	—	4	1	no	—
298	C₂×C₆₇	#2317	1	1	yes	characteristic
299	C₂×C₆₇	#2916	1	1	yes	characteristic
300	C₂×C₆₇	#2916	1	1	yes	characteristic
301	#2921	—	67	1	no	—
302	C₂×#433	—	67	1	no	—
303	#2905	—	67	1	no	—
304	#1950	—	67	1	no	—
305	#1950	—	67	1	no	—
306	#1950	—	67	1	no	—
307	#1950	—	67	1	no	—
308	#1855	—	67	1	no	—
309	#1855	—	67	1	no	—
310	#1855	—	67	1	no	—
311	#1855	D₆₇	1	1	yes	characteristic
312	#1849	—	67	1	no	—
313	#1848	—	67	1	no	—
314	#1838	—	67	1	no	—
315	#1837	—	67	1	no	—
316	C₆₇	#21995	1	1	yes	characteristic, complemented, Sylow
317	C₂²×M₁₆	—	67	1	no	—
318	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
319	C₂×#117	—	67	1	no	—
320	#433	—	67	2	no	—
321	#433	—	67	2	no	—
322	#433	—	67	2	no	—
323	#433	—	67	2	no	—
324	#433	—	134	1	no	—
325	#433	—	134	1	no	—
326	#433	—	134	1	no	—
327	#433	—	134	1	no	—
328	C₂×#108	—	67	1	no	—
329	C₂×#108	—	67	1	no	—
330	C₂×#108	—	67	1	no	—
331	C₂×#108	—	67	1	no	—
332	C₂×#108	—	67	1	no	—
333	C₂×#108	—	67	1	no	—
334	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
335	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
336	#424	—	134	1	no	—
337	#424	—	134	1	no	—
338	#424	—	134	1	no	—
339	#424	—	134	1	no	—
340	#423	—	67	1	no	—
341	#423	—	67	1	no	—
342	#423	—	67	1	no	—
343	#423	—	67	1	no	—
344	#423	—	67	1	no	—
345	#423	—	67	1	no	—
346	C₂×C₄⋊C₈	—	67	1	no	—
347	C₂×C₄⋊C₈	—	67	1	no	—
348	#407	—	67	1	no	—
349	#407	C₂×D₆₇	1	1	yes	characteristic
350	C₂×C₂²⋊C₈	—	67	1	no	—
351	C₂×C₂²⋊C₈	C₂×D₆₇	1	1	yes	characteristic
352	C₄×M₁₆	—	67	1	no	—
353	C₄×M₁₆	—	67	1	no	—
354	C₂×#97	—	67	1	no	—
355	C₂×C₄×C₈	—	67	1	no	—
356	#341	—	67	1	no	—
357	#341	—	67	1	no	—
358	#341	—	67	1	no	—
359	#341	—	67	1	no	—
360	#341	—	67	1	no	—
361	#341	—	67	1	no	—
362	#341	C₂×D₆₇	1	1	yes	characteristic
363	#341	C₂×D₆₇	1	1	yes	characteristic
364	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
365	C₂×M₁₆	—	2	1	no	—
366	C₂×M₁₆	—	4	1	no	—
367	C₂×M₁₆	—	4	1	no	—
368	C₂×M₁₆	—	268	1	no	—
369	C₂×M₁₆	—	268	1	no	—
370	C₂×M₁₆	—	67	1	no	—
371	C₂×M₁₆	—	67	1	no	—
372	C₂×M₁₆	—	67	1	no	—
373	C₂×M₁₆	—	67	1	no	—
374	C₂×M₁₆	—	134	1	no	—
375	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
376	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
377	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
378	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
379	C₂²×C₈	—	2	1	no	—
380	C₂²×C₈	—	67	1	no	—
381	C₂²×C₈	—	67	1	no	—
382	C₂²×C₈	—	134	1	no	—
383	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
384	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
385	#117	—	67	1	no	—
386	#117	—	67	1	no	—
387	#117	—	67	1	no	—
388	#117	—	67	1	no	—
389	#117	—	67	1	no	—
390	#117	—	67	1	no	—
391	#117	—	67	1	no	—
392	#117	—	67	1	no	—
393	C₂×C₄⋊C₄	—	67	1	no	—
394	C₂×C₄⋊C₄	—	67	1	no	—
395	C₂×C₂²⋊C₄	—	67	1	no	—
396	C₂×C₂²⋊C₄	—	67	1	no	—
397	C₂×C₄²	—	67	1	no	—
398	C₂×C₄²	—	67	1	no	—
399	#108	—	4	1	no	—
400	#108	—	4	1	no	—
401	#108	—	268	1	no	—
402	#108	—	268	1	no	—
403	#108	—	268	1	no	—
404	#108	—	268	1	no	—
405	#108	—	268	1	no	—
406	#108	—	268	1	no	—
407	C₄⋊C₈	—	67	2	no	—
408	C₄⋊C₈	—	67	2	no	—
409	C₄⋊C₈	—	67	2	no	—
410	C₄⋊C₈	—	67	2	no	—
411	C₄⋊C₈	—	134	1	no	—
412	C₄⋊C₈	—	134	1	no	—
413	C₄⋊C₈	—	134	1	no	—
414	C₄⋊C₈	—	134	1	no	—
415	C₄⋊C₈	—	134	1	no	—
416	C₄⋊C₈	—	134	1	no	—
417	C₄⋊C₈	—	134	1	no	—
418	C₄⋊C₈	—	134	1	no	—
419	C₄⋊C₈	—	134	1	no	—
420	C₄⋊C₈	—	134	1	no	—
421	C₄⋊C₈	—	134	1	no	—
422	C₄⋊C₈	—	134	1	no	—
423	C₂²⋊C₈	—	2	1	no	—
424	C₂²⋊C₈	—	2	1	no	—
425	C₂²⋊C₈	—	67	2	no	—
426	C₂²⋊C₈	—	67	2	no	—
427	C₂²⋊C₈	—	134	1	no	—
428	C₂²⋊C₈	—	134	1	no	—
429	C₂²⋊C₈	C₄×D₆₇	1	2	yes	—
430	C₂²⋊C₈	C₄×D₆₇	1	2	yes	—
431	#97	—	67	2	no	—
432	#97	—	67	2	no	—
433	#97	—	134	1	no	—
434	#97	—	134	1	no	—
435	C₄×C₈	—	67	2	no	—
436	C₄×C₈	—	67	2	no	—
437	C₄×C₈	—	134	1	no	—
438	C₄×C₈	—	134	1	no	—
439	C₂⁴	Q₈×D₆₇	1	1	yes	characteristic
440	C₂²×C₄	—	2	1	no	—
441	C₂²×C₄	—	2	1	no	—
442	C₂²×C₄	—	2	1	no	—
443	C₂²×C₄	—	2	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	C₂×C₄×D₆₇	1	1	yes	characteristic
449	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
450	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
451	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
452	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
453	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
454	M₁₆	—	2	1	no	—
455	M₁₆	—	2	1	no	—
456	M₁₆	—	2	1	no	—
457	M₁₆	—	2	1	no	—
458	M₁₆	—	2	1	no	—
459	M₁₆	—	2	1	no	—
460	M₁₆	—	2	1	no	—
461	M₁₆	—	2	1	no	—
462	M₁₆	—	4	2	no	—
463	M₁₆	—	268	2	no	—
464	M₁₆	—	134	1	no	—
465	M₁₆	—	134	1	no	—
466	M₁₆	—	134	1	no	—
467	M₁₆	—	134	1	no	—
468	M₁₆	—	134	1	no	—
469	M₁₆	—	134	1	no	—
470	M₁₆	—	134	1	no	—
471	M₁₆	—	134	1	no	—
472	C₂×C₈	—	2	1	no	—
473	C₂×C₈	—	2	1	no	—
474	C₂×C₈	—	2	1	no	—
475	C₂×C₈	—	2	1	no	—
476	C₂×C₈	—	2	1	no	—
477	C₂×C₈	—	2	1	no	—
478	C₂×C₈	—	2	1	no	—
479	C₂×C₈	—	2	1	no	—
480	C₂×C₈	—	4	2	no	—
481	C₂×C₈	—	268	2	no	—
482	C₂×C₈	—	67	2	no	—
483	C₂×C₈	—	67	2	no	—
484	C₂×C₈	—	134	1	no	—
485	C₂×C₈	—	134	1	no	—
486	C₂×C₈	—	134	1	no	—
487	C₂×C₈	—	134	1	no	—
488	C₂×C₈	—	134	1	no	—
489	C₂×C₈	—	134	1	no	—
490	C₂×C₈	—	134	1	no	—
491	C₂×C₈	—	134	1	no	—
492	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
493	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
494	C₄⋊C₄	—	67	1	no	—
495	C₄⋊C₄	—	67	1	no	—
496	C₄⋊C₄	—	67	1	no	—
497	C₄⋊C₄	—	67	1	no	—
498	C₄⋊C₄	—	67	1	no	—
499	C₄⋊C₄	—	67	1	no	—
500	C₄⋊C₄	—	67	1	no	—
501	C₄⋊C₄	—	67	1	no	—
502	C₂²⋊C₄	—	67	1	no	—
503	C₂²⋊C₄	—	67	1	no	—
504	C₂²⋊C₄	—	67	1	no	—
505	C₂²⋊C₄	—	67	1	no	—
506	C₂²⋊C₄	—	67	1	no	—
507	C₂²⋊C₄	—	67	1	no	—
508	C₂²⋊C₄	—	67	1	no	—
509	C₂²⋊C₄	—	67	1	no	—
510	C₄²	—	67	1	no	—
511	C₄²	—	67	1	no	—
512	C₄²	—	67	1	no	—
513	C₄²	—	67	1	no	—
514	C₄²	—	67	1	no	—
515	C₄²	—	67	1	no	—
516	C₄²	—	67	1	no	—
517	C₄²	—	67	1	no	—
518	C₂³	—	2	1	no	—
519	C₂³	—	2	1	no	—
520	C₂³	—	4	1	no	—
521	C₂³	—	4	1	no	—
522	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
523	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
524	C₂³	C₂×Q₈×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	1	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	268	1	no	—
536	C₂×C₄	—	268	1	no	—
537	C₂×C₄	—	268	1	no	—
538	C₂×C₄	—	268	1	no	—
539	C₂×C₄	—	67	1	no	—
540	C₂×C₄	—	67	1	no	—
541	C₂×C₄	—	67	1	no	—
542	C₂×C₄	—	67	1	no	—
543	C₂×C₄	—	67	1	no	—
544	C₂×C₄	—	67	1	no	—
545	C₂×C₄	—	67	1	no	—
546	C₂×C₄	—	67	1	no	—
547	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
548	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
549	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
550	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
551	C₂×C₄	#14524088	1	1	yes	characteristic
552	C₂×C₄	#14524088	1	1	yes	characteristic
553	C₂×C₄	#14524090	1	1	yes	characteristic
554	C₂×C₄	#14524090	1	1	yes	characteristic
555	C₈	—	2	2	no	—
556	C₈	—	2	2	no	—
557	C₈	—	4	1	no	—
558	C₈	—	4	1	no	—
559	C₈	—	4	1	no	—
560	C₈	—	4	1	no	—
561	C₈	—	268	1	no	—
562	C₈	—	268	1	no	—
563	C₈	—	268	1	no	—
564	C₈	—	268	1	no	—
565	C₈	—	134	2	no	—
566	C₈	—	134	2	no	—
567	C₂²	—	2	1	no	—
568	C₂²	—	2	1	no	—
569	C₂²	—	2	1	no	—
570	C₂²	—	2	1	no	—
571	C₂²	—	2	1	no	—
572	C₂²	—	2	1	no	—
573	C₂²	—	2	1	no	—
574	C₂²	—	2	1	no	—
575	C₂²	—	8	1	no	—
576	C₂²	—	8	1	no	—
577	C₂²	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
578	C₂²	#16864025	1	1	yes	characteristic
579	C₂²	#16864025	1	1	yes	characteristic
580	C₄	—	2	1	no	—
581	C₄	—	2	1	no	—
582	C₄	—	134	1	no	—
583	C₄	—	134	1	no	—
584	C₄	—	134	1	no	—
585	C₄	—	134	1	no	—
586	C₄	—	134	1	no	—
587	C₄	—	134	1	no	—
588	C₄	—	134	1	no	—
589	C₄	—	134	1	no	—
590	C₄	#16863875	1	1	yes	characteristic
591	C₄	#16863875	1	1	yes	characteristic
592	C₄	#16863961	1	1	yes	characteristic
593	C₄	#16863968	1	1	yes	characteristic
594	C₂	—	2	1	no	—
595	C₂	—	2	1	no	—
596	C₂	—	4	1	no	—
597	C₂	—	4	1	no	—
598	C₂	#22562828	1	1	yes	characteristic
599	C₂	#22568817	1	1	yes	characteristic
600	C₂	#22568819	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1855	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₆₇	C₂×Q₈
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1855	D₆₇
hypocenter	C₆₇	#21995
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1855

Complements

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

N	H ≅ G/N	type
4. #22560983	C₂	semidirect
5. #22560983	C₂	semidirect
6. #22560981	C₂	semidirect
7. #22560981	C₂	semidirect
11. #22560382	C₂	semidirect
12. #22560379	C₂	semidirect
13. #22560348	C₂	semidirect
14. #22560345	C₂	semidirect
316. C₆₇	#21995	semidirect

Maximal subgroups

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

subgroup	index
#22568844	2
C₂×#16863454	2
#22568772	2
#22560983	2
#22560981	2
#22560416	2
#22560414	2
#22560413	2
#22560382	2
#22560379	2
#22560348	2
#22560345	2
C₆₇×#1855	2
#21995	67