Subgroups — #30824927

Order 17152.

702 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568839	C₂	1	1	yes	maximal, characteristic
2	#22568837	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₈×D₆₇	C₂	1	1	yes	maximal, characteristic
4	#22558995	C₂	1	1	yes	maximal, characteristic, complemented
5	#22558994	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558992	C₂	1	1	yes	maximal, characteristic, complemented
7	#22558982	C₂	1	2	yes	maximal, complemented
8	#22558978	C₂	1	2	yes	maximal, complemented
9	#22558960	C₂	1	2	yes	maximal, complemented
10	#22558958	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1573	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₈×D₆₇	—	2	1	no	—
13	C₂×C₄²×D₆₇	C₂²	1	1	yes	characteristic
14	#16863453	C₂²	1	1	yes	characteristic
15	#16863452	C₄	1	2	yes	—
16	#16863451	—	2	1	no	—
17	#16863450	C₄	1	2	yes	—
18	#16863449	C₂²	1	1	yes	characteristic
19	#16863448	C₂²	1	1	yes	characteristic
20	#16863448	C₂²	1	1	yes	characteristic
21	#16863447	C₂²	1	2	yes	—
22	#16863446	—	2	1	no	—
23	#16863418	C₄	1	2	yes	—
24	#16863418	C₂²	1	1	yes	characteristic
25	#16863418	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
27	C₂×Dic₆₇⋊C₈	—	2	1	no	—
28	C₄⋊C₈×D₆₇	—	2	1	no	—
29	C₄⋊C₈×D₆₇	—	2	1	no	—
30	C₄⋊C₈×D₆₇	C₂²	1	2	yes	—
31	C₄⋊C₈×D₆₇	C₂²	1	2	yes	—
32	#16863358	—	2	1	no	—
33	#16863358	C₄	1	2	yes	—
34	#16863348	—	2	1	no	—
35	C₄×#11761611	C₂²	1	2	yes	—
36	#16863149	C₂²	1	2	yes	complemented
37	#16863148	C₂²	1	2	yes	complemented
38	#16863148	C₂²	1	2	yes	complemented
39	#16863147	C₂²	1	2	yes	complemented
40	#16863146	C₂²	1	2	yes	complemented
41	#16863146	C₂²	1	2	yes	complemented
42	#432×C₆₇	C₂²	1	1	yes	characteristic
43	#431×C₆₇	C₂²	1	1	yes	characteristic
44	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
45	#335×C₆₇	C₂²	1	2	yes	complemented
46	#334×C₆₇	C₂²	1	2	yes	complemented
47	C₂²×C₄×D₆₇	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×D₆₇	D₄	1	1	yes	characteristic
49	C₂²×C₄×D₆₇	Q₈	1	1	yes	characteristic
50	C₂×#11761611	—	2	1	no	—
51	C₂×#11761611	—	2	1	no	—
52	C₂×#11761611	C₂×C₄	1	4	yes	—
53	C₂²×C₆₇⋊C₈	—	2	1	no	—
54	C₂×C₈×D₆₇	—	2	2	no	—
55	C₂×C₈×D₆₇	—	2	2	no	—
56	C₂×C₈×D₆₇	—	4	4	no	—
57	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
59	C₂×C₄×Dic₆₇	Q₈	1	1	yes	characteristic
60	C₄²×D₆₇	C₂×C₄	1	1	yes	characteristic
61	C₄²×D₆₇	C₂×C₄	1	1	yes	characteristic
62	C₄²×D₆₇	D₄	1	1	yes	characteristic
63	C₄²×D₆₇	D₄	1	1	yes	characteristic
64	C₄²×D₆₇	Q₈	1	1	yes	characteristic
65	C₄²×D₆₇	Q₈	1	1	yes	characteristic
66	C₄²×D₆₇	C₂³	1	1	yes	characteristic
67	C₄²×D₆₇	C₂³	1	1	yes	characteristic
68	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
69	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
70	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
71	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
72	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
73	Dic₆₇⋊C₈	—	2	2	no	—
74	Dic₆₇⋊C₈	—	2	2	no	—
75	Dic₆₇⋊C₈	—	2	1	no	—
76	Dic₆₇⋊C₈	—	2	1	no	—
77	Dic₆₇⋊C₈	C₂×C₄	1	4	yes	—
78	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
79	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
80	(C₂×D₆₇)⋊C₈	—	2	1	no	—
81	(C₂×D₆₇)⋊C₈	—	2	1	no	—
82	(C₂×D₆₇)⋊C₈	C₂×C₄	1	4	yes	—
83	#14523954	—	2	1	no	—
84	#14523953	C₂×C₄	1	4	yes	—
85	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
86	C₈×Dic₆₇	—	2	1	no	—
87	C₈×Dic₆₇	—	2	1	no	—
88	C₂²×C₈×C₆₇	—	2	1	no	—
89	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
90	C₄⋊C₈×C₆₇	—	2	1	no	—
91	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
92	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
93	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
94	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
95	C₂²⋊C₈×C₆₇	—	2	1	no	—
96	C₂²⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
97	#97×C₆₇	—	2	1	no	—
98	C₄×C₈×C₆₇	C₂³	1	2	yes	—
99	C₂³×D₆₇	C₄⋊C₄	1	1	yes	characteristic
100	C₂²×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
101	C₂²×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
102	C₂²×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×D₆₇	—	2	4	no	—
104	C₂×C₄×D₆₇	—	2	2	no	—
105	C₂×C₄×D₆₇	—	2	2	no	—
106	C₂×C₄×D₆₇	—	4	2	no	—
107	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×D₆₇	C₂×Q₈	1	1	yes	characteristic
118	C₂×C₄×D₆₇	C₂×Q₈	1	1	yes	characteristic
119	#11761611	—	2	8	no	—
120	#11761611	—	4	2	no	—
121	#11761611	—	4	2	no	—
122	C₂×C₆₇⋊C₈	—	2	1	no	—
123	C₂×C₆₇⋊C₈	—	2	1	no	—
124	C₂×C₆₇⋊C₈	—	2	2	no	—
125	C₂×C₆₇⋊C₈	—	4	2	no	—
126	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
127	C₈×D₆₇	—	4	4	no	—
128	C₈×D₆₇	—	4	4	no	—
129	C₄×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
130	C₄×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
131	C₄×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
132	C₄×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
133	C₄×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
134	C₄×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
135	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
136	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
137	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
138	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
139	C₄×Dic₆₇	C₂×Q₈	1	1	yes	characteristic
140	C₄×Dic₆₇	C₂×Q₈	1	1	yes	characteristic
141	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
142	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
144	C₂×C₈×C₆₇	—	2	2	no	—
145	C₂×C₈×C₆₇	—	2	1	no	—
146	C₂×C₈×C₆₇	—	2	1	no	—
147	C₂×C₈×C₆₇	—	4	2	no	—
148	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
149	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
150	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
151	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
152	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
153	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₂²×D₆₇	—	2	2	no	—
155	C₂²×D₆₇	—	2	2	no	—
156	C₂²×D₆₇	—	2	2	no	—
157	C₂²×D₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
158	C₂²×D₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
159	C₂×C₄×C₆₇	—	2	2	no	—
160	C₂×C₄×C₆₇	—	2	2	no	—
161	C₂×C₄×C₆₇	—	2	1	no	—
162	C₂×C₄×C₆₇	—	2	1	no	—
163	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
165	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
166	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
167	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
169	C₂×Dic₆₇	—	2	2	no	—
170	C₂×Dic₆₇	—	2	2	no	—
171	C₂×Dic₆₇	—	2	2	no	—
172	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
173	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
174	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
175	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
176	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
177	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
178	C₄×D₆₇	—	2	2	no	—
179	C₄×D₆₇	—	2	2	no	—
180	C₄×D₆₇	—	2	2	no	—
181	C₄×D₆₇	—	2	1	no	—
182	C₄×D₆₇	—	2	1	no	—
183	C₄×D₆₇	—	2	2	no	—
184	C₄×D₆₇	—	2	2	no	—
185	C₄×D₆₇	—	4	1	no	—
186	C₄×D₆₇	—	4	1	no	—
187	C₄×D₆₇	—	4	1	no	—
188	C₄×D₆₇	—	4	1	no	—
189	C₄×D₆₇	#131	1	4	yes	—
190	C₄×D₆₇	AGL(1,ℤ/8)	1	2	yes	—
191	C₄×D₆₇	#137	1	2	yes	—
192	C₈×C₆₇	—	2	4	no	—
193	C₈×C₆₇	—	4	2	no	—
194	C₈×C₆₇	—	4	1	no	—
195	C₈×C₆₇	—	4	1	no	—
196	C₆₇⋊C₈	—	2	4	no	—
197	C₆₇⋊C₈	—	4	2	no	—
198	C₆₇⋊C₈	—	4	1	no	—
199	C₆₇⋊C₈	—	4	1	no	—
200	C₂²×C₆₇	—	2	1	no	—
201	C₂²×C₆₇	—	2	1	no	—
202	C₂²×C₆₇	#430	1	2	yes	—
203	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
204	C₂×D₆₇	—	2	8	no	—
205	C₂×D₆₇	—	2	1	no	—
206	C₂×D₆₇	—	2	1	no	—
207	C₂×D₆₇	—	2	1	no	—
208	C₂×D₆₇	—	2	1	no	—
209	C₂×D₆₇	#424	1	2	yes	—
210	C₂×D₆₇	#428	1	2	yes	—
211	C₄×C₆₇	—	2	1	no	—
212	C₄×C₆₇	—	2	1	no	—
213	C₄×C₆₇	—	2	1	no	—
214	C₄×C₆₇	—	2	1	no	—
215	C₄×C₆₇	C₂×#131	1	2	yes	—
216	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
217	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
218	Dic₆₇	—	2	1	no	—
219	Dic₆₇	—	2	1	no	—
220	Dic₆₇	—	2	1	no	—
221	Dic₆₇	—	2	1	no	—
222	Dic₆₇	#424	1	2	yes	—
223	Dic₆₇	#428	1	2	yes	—
224	#17807	—	67	1	no	maximal, Sylow
225	C₂×C₆₇	—	2	2	no	—
226	C₂×C₆₇	C₂×#424	1	1	yes	characteristic
227	C₂×C₆₇	C₂×#428	1	1	yes	characteristic
228	C₂×C₆₇	C₂×#430	1	1	yes	characteristic
229	D₆₇	—	2	2	no	—
230	D₆₇	—	2	2	no	—
231	#2920	—	67	1	no	—
232	#2920	—	67	1	no	—
233	C₂²×C₄⋊C₈	—	67	1	no	—
234	#1573	—	67	1	no	—
235	#1573	—	67	1	no	—
236	#1573	—	67	1	no	—
237	#1573	D₆₇	1	1	yes	characteristic, complemented
238	#1571	—	67	2	no	—
239	#1570	—	67	2	no	—
240	#1565	—	67	2	no	—
241	#1565	—	67	2	no	—
242	C₆₇	#17807	1	1	yes	characteristic, complemented, Sylow
243	C₂³×C₈	—	134	1	no	—
244	C₂²×C₄²	—	67	1	no	—
245	#432	—	67	2	no	—
246	#432	—	67	2	no	—
247	#432	—	67	1	no	—
248	#432	—	134	1	no	—
249	#432	C₂×D₆₇	1	1	yes	characteristic
250	#431	—	67	1	no	—
251	#431	—	67	2	no	—
252	#431	—	67	1	no	—
253	#431	—	67	1	no	—
254	#431	—	134	1	no	—
255	#431	C₂×D₆₇	1	1	yes	characteristic
256	#423	—	67	2	no	—
257	#423	—	67	1	no	—
258	#423	—	67	1	no	—
259	C₂×C₄⋊C₈	—	67	1	no	—
260	C₂×C₄⋊C₈	—	67	2	no	—
261	C₂×C₄⋊C₈	—	67	2	no	—
262	C₂×C₄⋊C₈	—	134	1	no	—
263	C₂×C₄⋊C₈	—	134	1	no	—
264	C₂×C₄⋊C₈	—	134	1	no	—
265	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
266	#407	—	67	2	no	—
267	#407	—	134	1	no	—
268	C₄×M₁₆	—	67	2	no	—
269	C₄×M₁₆	—	134	1	no	—
270	#335	—	67	2	no	—
271	#335	—	67	2	no	—
272	#335	—	67	2	no	—
273	#335	C₂×D₆₇	1	2	yes	complemented
274	#334	—	67	2	no	—
275	#334	—	67	2	no	—
276	#334	—	67	2	no	—
277	#334	C₂×D₆₇	1	2	yes	complemented
278	C₂³×C₄	—	67	1	no	—
279	C₂³×C₄	—	67	1	no	—
280	C₂³×C₄	—	67	1	no	—
281	C₂×M₁₆	—	67	4	no	—
282	C₂×M₁₆	—	134	1	no	—
283	C₂×M₁₆	—	134	1	no	—
284	C₂²×C₈	—	2	1	no	—
285	C₂²×C₈	—	268	4	no	—
286	C₂²×C₈	—	134	2	no	—
287	C₂²×C₈	—	134	2	no	—
288	C₂²×C₈	—	134	1	no	—
289	C₂×C₄²	—	67	1	no	—
290	C₂×C₄²	—	67	1	no	—
291	C₂×C₄²	—	67	1	no	—
292	C₂×C₄²	—	67	1	no	—
293	C₂×C₄²	—	67	1	no	—
294	C₂×C₄²	—	67	1	no	—
295	C₂×C₄²	—	67	1	no	—
296	C₂×C₄²	—	67	1	no	—
297	C₂×C₄²	—	67	1	no	—
298	C₂×C₄²	—	67	1	no	—
299	C₂×C₄²	—	67	1	no	—
300	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
301	C₄⋊C₈	—	2	1	no	—
302	C₄⋊C₈	—	67	4	no	—
303	C₄⋊C₈	—	67	2	no	—
304	C₄⋊C₈	—	67	1	no	—
305	C₄⋊C₈	—	67	2	no	—
306	C₄⋊C₈	—	67	1	no	—
307	C₄⋊C₈	—	134	2	no	—
308	C₄⋊C₈	—	134	2	no	—
309	C₄⋊C₈	—	134	1	no	—
310	C₄⋊C₈	—	134	1	no	—
311	C₄⋊C₈	—	134	1	no	—
312	C₄⋊C₈	C₄×D₆₇	1	2	yes	—
313	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
314	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
315	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
316	C₂²⋊C₈	—	2	1	no	—
317	C₂²⋊C₈	—	67	2	no	—
318	C₂²⋊C₈	—	67	4	no	—
319	C₂²⋊C₈	—	134	1	no	—
320	C₂²⋊C₈	—	134	1	no	—
321	C₂²⋊C₈	—	134	1	no	—
322	C₂²⋊C₈	C₄×D₆₇	1	2	yes	—
323	#97	—	2	1	no	—
324	#97	—	67	4	no	—
325	#97	—	134	1	no	—
326	C₄×C₈	—	67	2	no	—
327	C₄×C₈	—	134	1	no	—
328	C₄×C₈	—	134	1	no	—
329	C₄×C₈	C₂²×D₆₇	1	2	yes	—
330	C₂⁴	—	67	1	no	—
331	C₂²×C₄	—	268	2	no	—
332	C₂²×C₄	—	67	1	no	—
333	C₂²×C₄	—	67	1	no	—
334	C₂²×C₄	—	67	1	no	—
335	C₂²×C₄	—	67	1	no	—
336	C₂²×C₄	—	67	1	no	—
337	C₂²×C₄	—	67	1	no	—
338	C₂²×C₄	—	67	1	no	—
339	C₂²×C₄	—	67	1	no	—
340	C₂²×C₄	—	67	1	no	—
341	C₂²×C₄	—	67	1	no	—
342	C₂²×C₄	—	67	1	no	—
343	C₂²×C₄	—	67	1	no	—
344	C₂²×C₄	—	67	1	no	—
345	C₂²×C₄	—	67	1	no	—
346	C₂²×C₄	—	67	1	no	—
347	C₂²×C₄	—	134	2	no	—
348	C₂²×C₄	—	134	4	no	—
349	C₂²×C₄	—	134	2	no	—
350	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
351	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
352	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
353	M₁₆	—	268	2	no	—
354	M₁₆	—	268	2	no	—
355	M₁₆	—	134	8	no	—
356	C₂×C₈	—	2	1	no	—
357	C₂×C₈	—	2	1	no	—
358	C₂×C₈	—	2	2	no	—
359	C₂×C₈	—	4	2	no	—
360	C₂×C₈	—	268	4	no	—
361	C₂×C₈	—	268	4	no	—
362	C₂×C₈	—	268	2	no	—
363	C₂×C₈	—	67	4	no	—
364	C₂×C₈	—	134	1	no	—
365	C₂×C₈	—	134	1	no	—
366	C₂×C₈	—	134	2	no	—
367	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
368	C₄²	—	67	1	no	—
369	C₄²	—	67	1	no	—
370	C₄²	—	67	1	no	—
371	C₄²	—	67	1	no	—
372	C₄²	—	67	1	no	—
373	C₄²	—	67	1	no	—
374	C₄²	—	67	1	no	—
375	C₄²	—	67	1	no	—
376	C₄²	—	67	1	no	—
377	C₄²	—	67	1	no	—
378	C₄²	—	67	1	no	—
379	C₄²	—	67	1	no	—
380	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
381	C₄²	D₄×D₆₇	1	1	yes	characteristic
382	C₄²	Q₈×D₆₇	1	1	yes	characteristic
383	C₄²	C₂³×D₆₇	1	1	yes	characteristic
384	C₂³	—	67	1	no	—
385	C₂³	—	67	1	no	—
386	C₂³	—	134	2	no	—
387	C₂³	—	134	2	no	—
388	C₂³	—	134	2	no	—
389	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	1	no	—
393	C₂×C₄	—	2	1	no	—
394	C₂×C₄	—	268	1	no	—
395	C₂×C₄	—	268	1	no	—
396	C₂×C₄	—	268	1	no	—
397	C₂×C₄	—	268	1	no	—
398	C₂×C₄	—	67	1	no	—
399	C₂×C₄	—	67	1	no	—
400	C₂×C₄	—	67	4	no	—
401	C₂×C₄	—	67	2	no	—
402	C₂×C₄	—	67	2	no	—
403	C₂×C₄	—	67	1	no	—
404	C₂×C₄	—	67	1	no	—
405	C₂×C₄	—	67	1	no	—
406	C₂×C₄	—	67	1	no	—
407	C₂×C₄	—	134	2	no	—
408	C₂×C₄	—	134	2	no	—
409	C₂×C₄	—	134	2	no	—
410	C₂×C₄	—	134	2	no	—
411	C₂×C₄	—	134	2	no	—
412	C₂×C₄	—	134	2	no	—
413	C₂×C₄	—	134	2	no	—
414	C₂×C₄	—	134	2	no	—
415	C₂×C₄	—	134	1	no	—
416	C₂×C₄	—	134	1	no	—
417	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
418	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
419	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
420	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
421	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
422	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
423	C₈	—	2	4	no	—
424	C₈	—	4	2	no	—
425	C₈	—	4	1	no	—
426	C₈	—	4	1	no	—
427	C₈	—	268	2	no	—
428	C₈	—	268	1	no	—
429	C₈	—	268	1	no	—
430	C₈	—	134	4	no	—
431	C₂²	—	2	1	no	—
432	C₂²	—	2	1	no	—
433	C₂²	—	67	2	no	—
434	C₂²	—	67	2	no	—
435	C₂²	—	134	8	no	—
436	C₂²	—	134	1	no	—
437	C₂²	—	134	1	no	—
438	C₂²	—	134	1	no	—
439	C₂²	—	134	1	no	—
440	C₂²	#16863443	1	2	yes	—
441	C₂²	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
442	C₄	—	2	1	no	—
443	C₄	—	2	1	no	—
444	C₄	—	2	1	no	—
445	C₄	—	2	1	no	—
446	C₄	—	67	2	no	—
447	C₄	—	67	2	no	—
448	C₄	—	134	1	no	—
449	C₄	—	134	1	no	—
450	C₄	—	134	1	no	—
451	C₄	—	134	1	no	—
452	C₄	#131×D₆₇	1	2	yes	—
453	C₄	AGL(1,ℤ/8)×D₆₇	1	1	yes	characteristic
454	C₄	#137×D₆₇	1	1	yes	characteristic
455	C₂	—	2	2	no	—
456	C₂	—	134	2	no	—
457	C₂	—	134	2	no	—
458	C₂	#424×D₆₇	1	1	yes	characteristic
459	C₂	#428×D₆₇	1	1	yes	characteristic
460	C₂	C₂×#16863443	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×#1573	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂³×D₆₇	C₄⋊C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1573	D₆₇
hypocenter	C₆₇	#17807
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1573

Complements

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

N	H ≅ G/N	type
4. #22558995	C₂	semidirect
5. #22558994	C₂	semidirect
6. #22558992	C₂	semidirect
7. #22558982	C₂	semidirect
8. #22558978	C₂	semidirect
9. #22558960	C₂	semidirect
10. #22558958	C₂	semidirect
11. C₆₇×#1573	C₂	semidirect
36. #16863149	C₂²	semidirect
37. #16863148	C₂²	semidirect
38. #16863148	C₂²	semidirect
39. #16863147	C₂²	semidirect
40. #16863146	C₂²	semidirect
41. #16863146	C₂²	semidirect
45. #335×C₆₇	C₂²	semidirect
46. #334×C₆₇	C₂²	semidirect
237. #1573	D₆₇	semidirect
242. C₆₇	#17807	semidirect
273. #335	C₂×D₆₇	semidirect
277. #334	C₂×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22568839	2
#22568837	2
C₂×C₄⋊C₈×D₆₇	2
#22558995	2
#22558994	2
#22558992	2
#22558982	2
#22558978	2
#22558960	2
#22558958	2
C₆₇×#1573	2
#17807	67