Subgroups — #30824876

Order 17152.

666 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569323	C₂	1	2	yes	maximal
2	C₄×#14524049	C₂	1	1	yes	maximal, characteristic
3	#22558975	C₂	1	2	yes	maximal, complemented
4	#22558974	C₂	1	2	yes	maximal, complemented
5	#22558972	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558971	C₂	1	1	yes	maximal, characteristic, complemented
7	#22558966	C₂	1	1	yes	maximal, characteristic, complemented
8	#22558960	C₂	1	2	yes	maximal, complemented
9	#22558956	C₂	1	1	yes	maximal, characteristic, complemented
10	#335×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1567	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524049	C₄	1	1	yes	characteristic
13	C₂×#14524049	C₄	1	1	yes	characteristic
14	C₄×#11761628	C₂²	1	1	yes	characteristic
15	#16863525	—	2	2	no	—
16	#16863472	—	2	2	no	—
17	#16863466	—	2	2	no	—
18	#16863465	—	2	2	no	—
19	#16863453	C₂²	1	2	yes	—
20	#16863452	C₂²	1	2	yes	—
21	#16863448	C₂²	1	2	yes	—
22	#16863418	C₂²	1	2	yes	—
23	C₄⋊C₈×D₆₇	C₂²	1	2	yes	—
24	#16863351	C₄	1	1	yes	characteristic
25	#16863351	C₄	1	1	yes	characteristic
26	#16863350	C₄	1	1	yes	characteristic
27	#16863350	C₄	1	1	yes	characteristic
28	#16863350	C₄	1	1	yes	characteristic
29	#16863350	C₄	1	1	yes	characteristic
30	C₄×#11761612	C₂²	1	1	yes	characteristic
31	C₄×#11761611	C₂²	1	1	yes	characteristic
32	C₄×#11761611	C₂²	1	1	yes	characteristic
33	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
34	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
35	#16863149	C₂²	1	1	yes	characteristic
36	#16863149	C₂²	1	1	yes	characteristic
37	#16863148	C₂²	1	2	yes	—
38	#16863148	C₂²	1	2	yes	—
39	#16863147	C₂²	1	2	yes	—
40	#16863146	C₂²	1	2	yes	—
41	#16863146	C₂²	1	2	yes	—
42	#423×C₆₇	C₂²	1	2	yes	—
43	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
44	#335×C₆₇	C₂²	1	1	yes	characteristic
45	#335×C₆₇	C₂²	1	1	yes	characteristic
46	#335×C₆₇	C₂²	1	2	yes	—
47	C₂×#11761628	C₂×C₄	1	1	yes	characteristic
48	#14524049	—	2	4	no	—
49	#14524049	—	2	4	no	—
50	C₂×#11761612	C₂×C₄	1	1	yes	characteristic
51	C₂×#11761612	C₂×C₄	1	1	yes	characteristic
52	C₂×#11761611	—	2	2	no	—
53	C₂×#11761611	C₂×C₄	1	1	yes	characteristic
54	C₂×#11761611	C₂×C₄	1	1	yes	characteristic
55	C₂×#11761611	C₂×C₄	1	1	yes	characteristic
56	C₂×#11761611	C₂×C₄	1	1	yes	characteristic
57	C₂×C₈×D₆₇	—	2	2	no	—
58	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
60	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
61	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
62	C₄×Dic₁₃₄	D₄	1	1	yes	characteristic
63	C₄×Dic₁₃₄	Q₈	1	1	yes	characteristic
64	C₄×D₂₆₈	D₄	1	1	yes	characteristic
65	C₄×D₂₆₈	Q₈	1	1	yes	characteristic
66	C₄×C₂²⋊D₆₇	D₄	1	1	yes	characteristic
67	C₄×C₂²⋊D₆₇	D₄	1	1	yes	characteristic
68	C₄×C₂²⋊D₆₇	Q₈	1	1	yes	characteristic
69	C₄×C₂²⋊D₆₇	Q₈	1	1	yes	characteristic
70	#14524002	C₂×C₄	1	1	yes	characteristic
71	#14524000	C₂×C₄	1	1	yes	characteristic
72	#14524000	C₂×C₄	1	1	yes	characteristic
73	C₄²×D₆₇	C₂³	1	1	yes	characteristic
74	C₄²×D₆₇	C₂³	1	1	yes	characteristic
75	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
76	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
77	Dic₆₇⋊C₈	—	2	2	no	—
78	Dic₆₇⋊C₈	—	2	2	no	—
79	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
80	(C₂×D₆₇)⋊C₈	—	2	2	no	—
81	(C₂×D₆₇)⋊C₈	—	2	2	no	—
82	#14523954	—	2	2	no	—
83	#14523954	C₂×C₄	1	1	yes	characteristic
84	#14523954	C₂×C₄	1	1	yes	characteristic
85	#14523953	—	2	2	no	—
86	#14523953	C₂×C₄	1	1	yes	characteristic
87	#14523953	C₂×C₄	1	1	yes	characteristic
88	#14523953	C₂×C₄	1	1	yes	characteristic
89	#14523953	C₂×C₄	1	1	yes	characteristic
90	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
91	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
92	C₈×Dic₆₇	—	2	2	no	—
93	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
94	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
95	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
96	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
97	C₂×M₁₆×C₆₇	—	2	2	no	—
98	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
99	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
100	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
101	C₄⋊C₈×C₆₇	—	2	2	no	—
102	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
103	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
104	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
105	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
106	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
107	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
108	#11761628	—	2	2	no	—
109	#11761628	—	2	2	no	—
110	C₂×Dic₁₃₄	C₄⋊C₄	1	1	yes	characteristic
111	C₂×D₂₆₈	C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₂²⋊D₆₇	C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₂²⋊D₆₇	C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×D₆₇	C₂×Q₈	1	1	yes	characteristic
119	C₂×C₄×D₆₇	C₂×Q₈	1	1	yes	characteristic
120	#11761612	—	2	2	no	—
121	#11761612	—	2	2	no	—
122	#11761611	—	2	2	no	—
123	#11761611	—	2	2	no	—
124	#11761611	—	2	2	no	—
125	#11761611	—	8	2	no	—
126	#11761611	M₁₆	1	4	yes	complemented
127	C₂×C₆₇⋊C₈	—	2	2	no	—
128	C₂×C₆₇⋊C₈	—	2	2	no	—
129	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
133	C₈×D₆₇	—	2	2	no	—
134	C₈×D₆₇	—	2	2	no	—
135	C₈×D₆₇	—	2	2	no	—
136	C₈×D₆₇	—	8	2	no	—
137	C₈×D₆₇	M₁₆	1	4	yes	complemented
138	C₄⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
139	C₄⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
140	Dic₆₇⋊C₄	C₄⋊C₄	1	1	yes	characteristic
141	Dic₆₇⋊C₄	C₄⋊C₄	1	1	yes	characteristic
142	Dic₆₇⋊C₄	C₄⋊C₄	1	1	yes	characteristic
143	Dic₆₇⋊C₄	C₄⋊C₄	1	1	yes	characteristic
144	C₂²⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
145	C₂²⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
146	(C₂×D₆₇)⋊C₄	C₄⋊C₄	1	1	yes	characteristic
147	(C₂×D₆₇)⋊C₄	C₄⋊C₄	1	1	yes	characteristic
148	(C₂×D₆₇)⋊C₄	C₄⋊C₄	1	1	yes	characteristic
149	(C₂×D₆₇)⋊C₄	C₄⋊C₄	1	1	yes	characteristic
150	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
151	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
152	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
153	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
154	C₄×Dic₆₇	C₂×Q₈	1	1	yes	characteristic
155	C₄×Dic₆₇	C₂×Q₈	1	1	yes	characteristic
156	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
157	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
159	M₁₆×C₆₇	—	8	2	no	—
160	C₂×C₈×C₆₇	—	2	2	no	—
161	C₂×C₈×C₆₇	—	2	2	no	—
162	C₂×C₈×C₆₇	—	2	2	no	—
163	C₂×C₈×C₆₇	—	2	2	no	—
164	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
165	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
166	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
167	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
168	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
169	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
170	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
171	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
172	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
173	C₂²×D₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
174	C₂²×D₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
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	—	4	1	no	—
180	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
181	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
182	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
183	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
184	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
185	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
186	C₂²⋊D₆₇	—	2	2	no	—
187	C₂²⋊D₆₇	—	2	2	no	—
188	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
189	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
190	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
191	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
192	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
193	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
194	D₂₆₈	—	2	2	no	—
195	C₄×D₆₇	—	2	1	no	—
196	C₄×D₆₇	—	2	1	no	—
197	C₄×D₆₇	—	2	1	no	—
198	C₄×D₆₇	—	2	1	no	—
199	C₄×D₆₇	—	2	1	no	—
200	C₄×D₆₇	—	2	1	no	—
201	C₄×D₆₇	—	4	1	no	—
202	C₄×D₆₇	—	4	1	no	—
203	C₄×D₆₇	C₂×M₁₆	1	2	yes	—
204	C₄×D₆₇	C₂×M₁₆	1	2	yes	—
205	Dic₁₃₄	—	2	2	no	—
206	C₈×C₆₇	—	2	1	no	—
207	C₈×C₆₇	—	2	1	no	—
208	C₈×C₆₇	—	4	2	no	—
209	C₈×C₆₇	—	4	2	no	—
210	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
211	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
212	C₆₇⋊C₈	—	2	1	no	—
213	C₆₇⋊C₈	—	2	1	no	—
214	C₆₇⋊C₈	—	4	2	no	—
215	C₆₇⋊C₈	—	4	2	no	—
216	C₆₇⋊C₈	C₂×M₁₆	1	2	yes	—
217	C₆₇⋊C₈	C₂×M₁₆	1	2	yes	—
218	C₂²×C₆₇	—	2	1	no	—
219	C₂²×C₆₇	—	2	1	no	—
220	C₂²×C₆₇	—	2	1	no	—
221	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
222	C₂×D₆₇	—	2	1	no	—
223	C₂×D₆₇	—	2	1	no	—
224	C₂×D₆₇	—	2	1	no	—
225	C₂×D₆₇	—	2	1	no	—
226	C₂×D₆₇	—	2	1	no	—
227	C₂×D₆₇	#423	1	2	yes	—
228	C₄×C₆₇	—	2	1	no	—
229	C₄×C₆₇	—	2	1	no	—
230	C₄×C₆₇	—	2	1	no	—
231	C₄×C₆₇	—	2	1	no	—
232	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
233	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
234	C₄×C₆₇	#576	1	1	yes	characteristic
235	C₄×C₆₇	#578	1	1	yes	characteristic
236	Dic₆₇	—	2	1	no	—
237	Dic₆₇	—	2	1	no	—
238	Dic₆₇	—	2	1	no	—
239	Dic₆₇	—	2	1	no	—
240	Dic₆₇	—	2	1	no	—
241	Dic₆₇	#423	1	2	yes	—
242	#17799	—	67	1	no	maximal, Sylow
243	C₂×C₆₇	—	4	1	no	—
244	C₂×C₆₇	C₂×#423	1	1	yes	characteristic
245	C₂×C₆₇	#2913	1	1	yes	characteristic
246	C₂×C₆₇	#2916	1	1	yes	characteristic
247	D₆₇	—	2	2	no	—
248	D₆₇	—	4	1	no	—
249	#2989	—	67	2	no	—
250	C₄×#131	—	67	1	no	—
251	#1569	—	67	2	no	—
252	#1569	—	67	2	no	—
253	#1568	—	67	1	no	—
254	#1568	—	67	1	no	—
255	#1567	—	67	1	no	—
256	#1567	D₆₇	1	1	yes	characteristic, complemented
257	#1565	—	67	2	no	—
258	#1564	—	67	1	no	—
259	C₂×#335	—	67	1	no	—
260	C₆₇	#17799	1	1	yes	characteristic, complemented, Sylow
261	C₂×#131	—	67	1	no	—
262	C₂×#131	—	67	1	no	—
263	C₄×#41	—	67	1	no	—
264	#446	—	134	2	no	—
265	#436	—	134	2	no	—
266	#435	—	134	2	no	—
267	#435	—	134	2	no	—
268	#432	—	67	2	no	—
269	#432	—	67	2	no	—
270	#431	—	67	2	no	—
271	#423	—	67	2	no	—
272	#423	C₂×D₆₇	1	2	yes	—
273	C₂×C₄⋊C₈	—	67	2	no	—
274	#405	—	67	1	no	—
275	#405	—	67	1	no	—
276	#405	—	67	1	no	—
277	#405	—	67	1	no	—
278	#405	—	67	1	no	—
279	#405	—	67	1	no	—
280	C₄×M₁₆	—	67	1	no	—
281	C₄×M₁₆	—	67	1	no	—
282	C₄×M₁₆	—	67	1	no	—
283	C₂×C₄×C₈	—	67	1	no	—
284	C₂×C₄×C₈	—	67	1	no	—
285	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
286	#335	—	67	2	no	—
287	#335	—	67	2	no	—
288	#335	—	67	1	no	—
289	#335	—	67	1	no	—
290	#335	C₂×D₆₇	1	1	yes	characteristic
291	#335	C₂×D₆₇	1	1	yes	characteristic
292	#335	C₂×D₆₇	1	2	yes	—
293	#334	—	67	2	no	—
294	#334	—	67	2	no	—
295	#334	—	67	2	no	—
296	C₂×#41	—	67	1	no	—
297	#131	—	134	4	no	—
298	#131	—	134	4	no	—
299	C₂×M₁₆	—	2	2	no	—
300	C₂×M₁₆	—	67	1	no	—
301	C₂×M₁₆	—	67	1	no	—
302	C₂×M₁₆	—	67	1	no	—
303	C₂×M₁₆	—	67	1	no	—
304	C₂×M₁₆	—	67	1	no	—
305	C₂×M₁₆	—	67	1	no	—
306	C₂×M₁₆	—	134	2	no	—
307	C₂²×C₈	—	67	1	no	—
308	C₂²×C₈	—	67	1	no	—
309	C₂²×C₈	—	67	1	no	—
310	C₂²×C₈	—	67	1	no	—
311	C₂²×C₈	—	134	2	no	—
312	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
313	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
314	C₄×Q₈	—	67	1	no	—
315	C₄×Q₈	—	67	1	no	—
316	C₄×D₄	—	67	1	no	—
317	C₄×D₄	—	67	1	no	—
318	C₄×D₄	—	67	1	no	—
319	C₄×D₄	—	67	1	no	—
320	C₄×D₄	—	67	1	no	—
321	C₄×D₄	—	67	1	no	—
322	#117	—	67	1	no	—
323	#117	—	67	1	no	—
324	#117	—	67	1	no	—
325	C₂×C₄²	—	67	1	no	—
326	C₂×C₄²	—	67	1	no	—
327	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
328	C₄⋊C₈	—	2	2	no	—
329	C₄⋊C₈	—	67	2	no	—
330	C₄⋊C₈	—	67	2	no	—
331	C₄⋊C₈	—	134	2	no	—
332	C₄⋊C₈	—	134	2	no	—
333	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
334	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
335	C₂²⋊C₈	—	134	2	no	—
336	C₂²⋊C₈	—	134	2	no	—
337	C₂²⋊C₈	—	134	2	no	—
338	#97	—	67	1	no	—
339	#97	—	67	1	no	—
340	#97	—	67	1	no	—
341	#97	—	67	1	no	—
342	#97	—	67	1	no	—
343	#97	—	67	1	no	—
344	#97	—	134	2	no	—
345	#97	—	134	2	no	—
346	C₄×C₈	—	67	1	no	—
347	C₄×C₈	—	67	1	no	—
348	C₄×C₈	—	67	1	no	—
349	C₄×C₈	—	67	1	no	—
350	C₄×C₈	—	67	1	no	—
351	C₄×C₈	—	67	1	no	—
352	C₄×C₈	—	134	2	no	—
353	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
354	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
355	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
356	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
357	#41	—	134	2	no	—
358	#41	—	134	2	no	—
359	C₂×Q₈	—	67	1	no	—
360	C₂×D₄	—	67	1	no	—
361	C₂×D₄	—	67	1	no	—
362	C₂×D₄	—	67	1	no	—
363	C₂²×C₄	—	67	1	no	—
364	C₂²×C₄	—	67	1	no	—
365	C₂²×C₄	—	67	1	no	—
366	C₂²×C₄	—	67	1	no	—
367	C₂²×C₄	—	67	1	no	—
368	C₂²×C₄	—	67	1	no	—
369	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
370	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
371	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
372	M₁₆	—	8	2	no	—
373	M₁₆	—	536	2	no	—
374	M₁₆	—	67	4	no	—
375	M₁₆	—	134	2	no	—
376	M₁₆	—	134	2	no	—
377	M₁₆	—	134	2	no	—
378	M₁₆	—	134	2	no	—
379	M₁₆	—	134	2	no	—
380	C₂×C₈	—	2	2	no	—
381	C₂×C₈	—	2	2	no	—
382	C₂×C₈	—	2	2	no	—
383	C₂×C₈	—	2	2	no	—
384	C₂×C₈	—	536	2	no	—
385	C₂×C₈	—	67	1	no	—
386	C₂×C₈	—	67	1	no	—
387	C₂×C₈	—	67	4	no	—
388	C₂×C₈	—	67	1	no	—
389	C₂×C₈	—	67	1	no	—
390	C₂×C₈	—	134	2	no	—
391	C₂×C₈	—	134	2	no	—
392	C₂×C₈	—	134	2	no	—
393	C₂×C₈	—	134	2	no	—
394	C₂×C₈	—	134	2	no	—
395	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
396	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
397	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
398	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
399	C₄⋊C₄	—	67	1	no	—
400	C₄⋊C₄	—	67	1	no	—
401	C₄⋊C₄	—	67	1	no	—
402	C₄⋊C₄	—	67	1	no	—
403	C₄⋊C₄	—	67	1	no	—
404	C₄⋊C₄	—	67	1	no	—
405	C₂²⋊C₄	—	67	1	no	—
406	C₂²⋊C₄	—	67	1	no	—
407	C₂²⋊C₄	—	67	1	no	—
408	C₂²⋊C₄	—	67	1	no	—
409	C₂²⋊C₄	—	67	1	no	—
410	C₂²⋊C₄	—	67	1	no	—
411	C₄²	—	67	1	no	—
412	C₄²	—	67	1	no	—
413	C₄²	—	67	1	no	—
414	C₄²	—	67	1	no	—
415	C₄²	—	67	1	no	—
416	C₄²	—	67	1	no	—
417	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
418	C₄²	D₄×D₆₇	1	1	yes	characteristic
419	C₄²	Q₈×D₆₇	1	1	yes	characteristic
420	C₄²	C₂³×D₆₇	1	1	yes	characteristic
421	C₂³	—	67	1	no	—
422	C₂³	—	67	1	no	—
423	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
424	Q₈	—	134	2	no	—
425	D₄	—	134	2	no	—
426	D₄	—	134	2	no	—
427	D₄	—	134	2	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	—	4	1	no	—
433	C₂×C₄	—	268	1	no	—
434	C₂×C₄	—	268	1	no	—
435	C₂×C₄	—	67	1	no	—
436	C₂×C₄	—	67	2	no	—
437	C₂×C₄	—	67	2	no	—
438	C₂×C₄	—	67	1	no	—
439	C₂×C₄	—	67	1	no	—
440	C₂×C₄	—	67	1	no	—
441	C₂×C₄	—	67	1	no	—
442	C₂×C₄	—	67	1	no	—
443	C₂×C₄	—	134	1	no	—
444	C₂×C₄	—	134	1	no	—
445	C₂×C₄	—	134	1	no	—
446	C₂×C₄	—	134	1	no	—
447	C₂×C₄	—	134	1	no	—
448	C₂×C₄	—	134	1	no	—
449	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
450	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
451	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
452	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
453	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
454	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
455	C₈	—	2	1	no	—
456	C₈	—	2	1	no	—
457	C₈	—	4	2	no	—
458	C₈	—	4	2	no	—
459	C₈	—	268	2	no	—
460	C₈	—	268	2	no	—
461	C₈	—	67	2	no	—
462	C₈	—	67	2	no	—
463	C₈	—	134	1	no	—
464	C₈	—	134	1	no	—
465	C₈	M₁₆×D₆₇	1	2	yes	—
466	C₈	M₁₆×D₆₇	1	2	yes	—
467	C₂²	—	2	1	no	—
468	C₂²	—	2	1	no	—
469	C₂²	—	2	1	no	—
470	C₂²	—	67	2	no	—
471	C₂²	—	134	1	no	—
472	C₂²	—	134	1	no	—
473	C₂²	—	134	1	no	—
474	C₂²	—	134	1	no	—
475	C₂²	—	134	1	no	—
476	C₂²	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
477	C₄	—	2	1	no	—
478	C₄	—	2	1	no	—
479	C₄	—	2	1	no	—
480	C₄	—	2	1	no	—
481	C₄	—	67	2	no	—
482	C₄	—	134	1	no	—
483	C₄	—	134	1	no	—
484	C₄	—	134	1	no	—
485	C₄	—	134	1	no	—
486	C₄	—	134	1	no	—
487	C₄	C₂×M₁₆×D₆₇	1	1	yes	characteristic
488	C₄	C₂×M₁₆×D₆₇	1	1	yes	characteristic
489	C₄	#16864197	1	1	yes	characteristic
490	C₄	#16864209	1	1	yes	characteristic
491	C₂	—	4	1	no	—
492	C₂	—	268	1	no	—
493	C₂	—	134	2	no	—
494	C₂	#423×D₆₇	1	1	yes	characteristic
495	C₂	#22568805	1	1	yes	characteristic
496	C₂	#22568817	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#1567	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂×#11761628	C₂×C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1567	D₆₇
hypocenter	C₆₇	#17799
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
3. #22558975	C₂	semidirect
4. #22558974	C₂	semidirect
5. #22558972	C₂	semidirect
6. #22558971	C₂	semidirect
7. #22558966	C₂	semidirect
8. #22558960	C₂	semidirect
9. #22558956	C₂	semidirect
10. #335×D₆₇	C₂	semidirect
11. C₆₇×#1567	C₂	semidirect
126. #11761611	M₁₆	semidirect
137. C₈×D₆₇	M₁₆	semidirect
256. #1567	D₆₇	semidirect
260. C₆₇	#17799	semidirect

Maximal subgroups

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

subgroup	index
#22569323	2
C₄×#14524049	2
#22558975	2
#22558974	2
#22558972	2
#22558971	2
#22558966	2
#22558960	2
#22558956	2
#335×D₆₇	2
C₆₇×#1567	2
#17799	67