Subgroups — #30824934

Order 17152.

646 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568841	C₂	1	1	yes	maximal, characteristic
2	#22568837	C₂	1	1	yes	maximal, characteristic
3	#431×D₆₇	C₂	1	1	yes	maximal, characteristic
4	#22559000	C₂	1	1	yes	maximal, characteristic, complemented
5	#22558998	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558997	C₂	1	1	yes	maximal, characteristic, complemented
7	#22558988	C₂	1	2	yes	maximal, complemented
8	#22558986	C₂	1	2	yes	maximal, complemented
9	#22558960	C₂	1	2	yes	maximal, complemented
10	#22558958	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1574	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄²×D₆₇	C₂²	1	1	yes	characteristic
13	#16863452	—	2	1	no	—
14	#16863452	C₄	1	2	yes	—
15	#16863452	C₂²	1	2	yes	—
16	#16863450	C₄	1	2	yes	—
17	#16863449	C₂²	1	1	yes	characteristic
18	#16863449	C₂²	1	1	yes	characteristic
19	#16863448	—	2	1	no	—
20	#16863448	C₂²	1	1	yes	characteristic
21	#16863448	C₂²	1	1	yes	characteristic
22	#16863448	C₂²	1	2	yes	—
23	#16863447	—	2	1	no	—
24	#16863447	C₂²	1	2	yes	—
25	#16863446	—	2	1	no	—
26	#16863420	C₂²	1	1	yes	characteristic
27	#16863418	C₄	1	2	yes	—
28	C₄⋊C₈×D₆₇	C₂²	1	1	yes	characteristic
29	C₄⋊C₈×D₆₇	C₂²	1	1	yes	characteristic
30	#16863359	—	2	1	no	—
31	#16863358	C₄	1	2	yes	—
32	C₂²⋊C₈×D₆₇	—	2	1	no	—
33	M₁₆×Dic₆₇	—	2	1	no	—
34	C₄×#11761611	C₂²	1	2	yes	—
35	C₄×C₈×D₆₇	—	2	1	no	—
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	#431×C₆₇	C₂²	1	1	yes	characteristic
43	#431×C₆₇	C₂²	1	1	yes	characteristic
44	#423×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×#11761612	—	2	1	no	—
51	C₂×#11761611	C₂×C₄	1	4	yes	—
52	C₂×C₈×D₆₇	—	2	1	no	—
53	C₂×C₈×D₆₇	—	2	1	no	—
54	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
56	C₂×C₄×Dic₆₇	Q₈	1	1	yes	characteristic
57	C₄²×D₆₇	C₂×C₄	1	1	yes	characteristic
58	C₄²×D₆₇	C₂×C₄	1	1	yes	characteristic
59	C₄²×D₆₇	D₄	1	1	yes	characteristic
60	C₄²×D₆₇	D₄	1	1	yes	characteristic
61	C₄²×D₆₇	Q₈	1	1	yes	characteristic
62	C₄²×D₆₇	Q₈	1	1	yes	characteristic
63	C₄²×D₆₇	C₂³	1	1	yes	characteristic
64	C₄²×D₆₇	C₂³	1	1	yes	characteristic
65	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
66	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
67	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
68	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
69	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
70	Dic₆₇⋊C₈	—	2	1	no	—
71	Dic₆₇⋊C₈	—	2	1	no	—
72	Dic₆₇⋊C₈	C₂×C₄	1	4	yes	—
73	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
74	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
75	(C₂×D₆₇)⋊C₈	—	2	1	no	—
76	(C₂×D₆₇)⋊C₈	—	2	1	no	—
77	(C₂×D₆₇)⋊C₈	—	2	2	no	—
78	(C₂×D₆₇)⋊C₈	—	2	2	no	—
79	(C₂×D₆₇)⋊C₈	C₂×C₄	1	4	yes	—
80	#14523953	—	2	2	no	—
81	#14523953	C₂×C₄	1	4	yes	—
82	C₄×C₆₇⋊C₈	—	2	1	no	—
83	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
84	C₈×Dic₆₇	—	2	1	no	—
85	C₈×Dic₆₇	—	2	1	no	—
86	C₈×Dic₆₇	—	2	2	no	—
87	C₂×M₁₆×C₆₇	—	2	1	no	—
88	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
89	C₄⋊C₈×C₆₇	—	2	1	no	—
90	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
91	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
92	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
93	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
94	C₂²⋊C₈×C₆₇	—	2	1	no	—
95	C₂²⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
96	C₄×C₈×C₆₇	—	2	1	no	—
97	C₄×C₈×C₆₇	C₂³	1	2	yes	—
98	C₂³×D₆₇	C₄⋊C₄	1	1	yes	characteristic
99	C₂²×Dic₆₇	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	C₂×C₄×D₆₇	—	2	4	no	—
103	C₂×C₄×D₆₇	—	4	1	no	—
104	C₂×C₄×D₆₇	—	4	1	no	—
105	C₂×C₄×D₆₇	—	4	2	no	—
106	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
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	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×D₆₇	C₂×Q₈	1	1	yes	characteristic
117	C₂×C₄×D₆₇	C₂×Q₈	1	1	yes	characteristic
118	#11761612	—	4	2	no	—
119	#11761611	—	2	8	no	—
120	C₂×C₆₇⋊C₈	—	2	1	no	—
121	C₂×C₆₇⋊C₈	—	2	1	no	—
122	C₂×C₆₇⋊C₈	—	2	2	no	—
123	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
124	C₈×D₆₇	—	4	2	no	—
125	C₈×D₆₇	—	4	2	no	—
126	C₄×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
127	C₄×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
128	C₄×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
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	C₂²×C₄	1	1	yes	characteristic
133	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
134	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
135	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
136	C₄×Dic₆₇	C₂×Q₈	1	1	yes	characteristic
137	C₄×Dic₆₇	C₂×Q₈	1	1	yes	characteristic
138	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
139	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
141	M₁₆×C₆₇	—	4	2	no	—
142	C₂×C₈×C₆₇	—	2	2	no	—
143	C₂×C₈×C₆₇	—	2	1	no	—
144	C₂×C₈×C₆₇	—	2	1	no	—
145	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
146	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
147	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
148	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
149	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
150	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂²×D₆₇	—	2	2	no	—
152	C₂²×D₆₇	—	4	1	no	—
153	C₂²×D₆₇	—	4	1	no	—
154	C₂²×D₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₂²×D₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
156	C₂×C₄×C₆₇	—	2	2	no	—
157	C₂×C₄×C₆₇	—	2	1	no	—
158	C₂×C₄×C₆₇	—	2	1	no	—
159	C₂×C₄×C₆₇	—	2	2	no	—
160	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
161	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
162	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
163	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
164	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
165	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
166	C₂×Dic₆₇	—	2	2	no	—
167	C₂×Dic₆₇	—	4	1	no	—
168	C₂×Dic₆₇	—	4	1	no	—
169	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
170	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
171	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
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	C₄×D₆₇	—	2	2	no	—
176	C₄×D₆₇	—	2	2	no	—
177	C₄×D₆₇	—	2	1	no	—
178	C₄×D₆₇	—	2	1	no	—
179	C₄×D₆₇	—	2	1	no	—
180	C₄×D₆₇	—	2	1	no	—
181	C₄×D₆₇	—	2	2	no	—
182	C₄×D₆₇	—	2	2	no	—
183	C₄×D₆₇	—	2	2	no	—
184	C₄×D₆₇	—	4	1	no	—
185	C₄×D₆₇	—	4	1	no	—
186	C₄×D₆₇	—	4	1	no	—
187	C₄×D₆₇	—	4	1	no	—
188	C₄×D₆₇	#131	1	4	yes	—
189	C₈×C₆₇	—	2	4	no	—
190	C₈×C₆₇	—	4	2	no	—
191	C₈×C₆₇	—	4	1	no	—
192	C₈×C₆₇	—	4	1	no	—
193	C₆₇⋊C₈	—	2	4	no	—
194	C₆₇⋊C₈	—	4	1	no	—
195	C₆₇⋊C₈	—	4	1	no	—
196	C₆₇⋊C₈	—	4	2	no	—
197	C₂²×C₆₇	—	2	1	no	—
198	C₂²×C₆₇	—	2	1	no	—
199	C₂²×C₆₇	—	2	1	no	—
200	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
201	C₂×D₆₇	—	2	1	no	—
202	C₂×D₆₇	—	2	1	no	—
203	C₂×D₆₇	—	2	1	no	—
204	C₂×D₆₇	—	2	1	no	—
205	C₂×D₆₇	—	2	1	no	—
206	C₂×D₆₇	—	4	4	no	—
207	C₂×D₆₇	#424	1	2	yes	—
208	C₄×C₆₇	—	2	1	no	—
209	C₄×C₆₇	—	2	1	no	—
210	C₄×C₆₇	—	2	1	no	—
211	C₄×C₆₇	#430	1	2	yes	—
212	C₄×C₆₇	C₂×#131	1	2	yes	—
213	C₄×C₆₇	#575	1	1	yes	characteristic
214	C₄×C₆₇	#575	1	1	yes	characteristic
215	Dic₆₇	—	2	1	no	—
216	Dic₆₇	—	2	1	no	—
217	Dic₆₇	—	2	1	no	—
218	Dic₆₇	—	2	1	no	—
219	Dic₆₇	—	2	1	no	—
220	Dic₆₇	#424	1	2	yes	—
221	#17808	—	67	1	no	maximal, Sylow
222	C₂×C₆₇	—	4	1	no	—
223	C₂×C₆₇	C₂×#424	1	1	yes	characteristic
224	C₂×C₆₇	#2911	1	1	yes	characteristic
225	C₂×C₆₇	C₂×#430	1	1	yes	characteristic
226	D₆₇	—	2	2	no	—
227	D₆₇	—	4	1	no	—
228	#2920	—	67	1	no	—
229	#2920	—	67	1	no	—
230	C₂×#431	—	67	1	no	—
231	#1574	—	67	1	no	—
232	#1574	—	67	1	no	—
233	#1574	—	67	1	no	—
234	#1574	D₆₇	1	1	yes	characteristic, complemented
235	#1572	—	67	2	no	—
236	#1572	—	67	2	no	—
237	#1565	—	67	2	no	—
238	#1565	—	67	2	no	—
239	C₆₇	#17808	1	1	yes	characteristic, complemented, Sylow
240	C₂²×C₄²	—	67	1	no	—
241	#432	—	67	2	no	—
242	#432	—	67	2	no	—
243	#432	—	67	2	no	—
244	#432	—	134	1	no	—
245	#431	—	67	1	no	—
246	#431	—	67	2	no	—
247	#431	—	67	1	no	—
248	#431	—	67	2	no	—
249	#431	—	67	1	no	—
250	#431	—	67	1	no	—
251	#431	—	134	1	no	—
252	#431	—	134	1	no	—
253	#431	—	134	1	no	—
254	#431	C₂×D₆₇	1	1	yes	characteristic
255	#431	C₂×D₆₇	1	1	yes	characteristic
256	#423	—	67	2	no	—
257	#423	—	67	1	no	—
258	#423	C₂×D₆₇	1	1	yes	characteristic
259	C₂×C₄⋊C₈	—	67	1	no	—
260	C₂×C₄⋊C₈	—	67	1	no	—
261	#407	—	67	2	no	—
262	#407	—	134	1	no	—
263	C₂×C₂²⋊C₈	—	134	1	no	—
264	C₄×M₁₆	—	67	2	no	—
265	C₄×M₁₆	—	134	1	no	—
266	C₂×C₄×C₈	—	134	1	no	—
267	#335	—	67	2	no	—
268	#335	—	67	2	no	—
269	#335	—	67	2	no	—
270	#335	C₂×D₆₇	1	2	yes	complemented
271	#334	—	67	2	no	—
272	#334	—	67	2	no	—
273	#334	—	67	2	no	—
274	#334	C₂×D₆₇	1	2	yes	complemented
275	C₂³×C₄	—	67	1	no	—
276	C₂³×C₄	—	67	1	no	—
277	C₂³×C₄	—	67	1	no	—
278	C₂×M₁₆	—	2	1	no	—
279	C₂×M₁₆	—	67	4	no	—
280	C₂×M₁₆	—	134	1	no	—
281	C₂²×C₈	—	134	1	no	—
282	C₂²×C₈	—	134	1	no	—
283	C₂×C₄²	—	67	1	no	—
284	C₂×C₄²	—	67	1	no	—
285	C₂×C₄²	—	67	1	no	—
286	C₂×C₄²	—	67	1	no	—
287	C₂×C₄²	—	67	1	no	—
288	C₂×C₄²	—	67	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	C₂²×D₆₇	1	1	yes	characteristic
295	C₄⋊C₈	—	2	1	no	—
296	C₄⋊C₈	—	67	4	no	—
297	C₄⋊C₈	—	67	2	no	—
298	C₄⋊C₈	—	67	1	no	—
299	C₄⋊C₈	—	67	1	no	—
300	C₄⋊C₈	—	67	2	no	—
301	C₄⋊C₈	—	134	1	no	—
302	C₄⋊C₈	—	134	1	no	—
303	C₄⋊C₈	—	134	1	no	—
304	C₄⋊C₈	C₄×D₆₇	1	2	yes	—
305	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
306	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
307	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
308	C₂²⋊C₈	—	2	1	no	—
309	C₂²⋊C₈	—	67	2	no	—
310	C₂²⋊C₈	—	67	4	no	—
311	C₂²⋊C₈	—	134	1	no	—
312	C₂²⋊C₈	—	134	1	no	—
313	C₂²⋊C₈	—	134	2	no	—
314	C₂²⋊C₈	—	134	2	no	—
315	C₂²⋊C₈	—	134	1	no	—
316	C₂²⋊C₈	C₄×D₆₇	1	2	yes	—
317	#97	—	67	4	no	—
318	#97	—	134	2	no	—
319	C₄×C₈	—	2	1	no	—
320	C₄×C₈	—	67	2	no	—
321	C₄×C₈	—	134	1	no	—
322	C₄×C₈	—	134	1	no	—
323	C₄×C₈	—	134	2	no	—
324	C₄×C₈	—	134	1	no	—
325	C₄×C₈	C₂²×D₆₇	1	2	yes	—
326	C₂⁴	—	67	1	no	—
327	C₂²×C₄	—	268	1	no	—
328	C₂²×C₄	—	268	1	no	—
329	C₂²×C₄	—	268	2	no	—
330	C₂²×C₄	—	67	1	no	—
331	C₂²×C₄	—	67	1	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	—	134	4	no	—
346	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
347	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
348	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
349	M₁₆	—	4	2	no	—
350	M₁₆	—	268	2	no	—
351	M₁₆	—	134	8	no	—
352	C₂×C₈	—	2	1	no	—
353	C₂×C₈	—	2	1	no	—
354	C₂×C₈	—	2	2	no	—
355	C₂×C₈	—	268	2	no	—
356	C₂×C₈	—	268	2	no	—
357	C₂×C₈	—	67	4	no	—
358	C₂×C₈	—	134	1	no	—
359	C₂×C₈	—	134	1	no	—
360	C₂×C₈	—	134	2	no	—
361	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
362	C₄²	—	67	1	no	—
363	C₄²	—	67	1	no	—
364	C₄²	—	67	1	no	—
365	C₄²	—	67	1	no	—
366	C₄²	—	67	1	no	—
367	C₄²	—	67	1	no	—
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	C₂×C₄×D₆₇	1	1	yes	characteristic
375	C₄²	D₄×D₆₇	1	1	yes	characteristic
376	C₄²	Q₈×D₆₇	1	1	yes	characteristic
377	C₄²	C₂³×D₆₇	1	1	yes	characteristic
378	C₂³	—	268	1	no	—
379	C₂³	—	268	1	no	—
380	C₂³	—	67	1	no	—
381	C₂³	—	67	1	no	—
382	C₂³	—	134	2	no	—
383	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	2	1	no	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	268	1	no	—
389	C₂×C₄	—	268	1	no	—
390	C₂×C₄	—	268	1	no	—
391	C₂×C₄	—	268	1	no	—
392	C₂×C₄	—	268	1	no	—
393	C₂×C₄	—	268	1	no	—
394	C₂×C₄	—	67	1	no	—
395	C₂×C₄	—	67	4	no	—
396	C₂×C₄	—	67	1	no	—
397	C₂×C₄	—	67	1	no	—
398	C₂×C₄	—	67	1	no	—
399	C₂×C₄	—	67	1	no	—
400	C₂×C₄	—	67	1	no	—
401	C₂×C₄	—	134	2	no	—
402	C₂×C₄	—	134	2	no	—
403	C₂×C₄	—	134	2	no	—
404	C₂×C₄	—	134	2	no	—
405	C₂×C₄	—	134	1	no	—
406	C₂×C₄	—	134	1	no	—
407	C₂×C₄	—	134	1	no	—
408	C₂×C₄	—	134	1	no	—
409	C₂×C₄	—	134	2	no	—
410	C₂×C₄	—	134	2	no	—
411	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
412	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
413	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
414	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
415	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
416	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
417	C₈	—	2	4	no	—
418	C₈	—	4	1	no	—
419	C₈	—	4	2	no	—
420	C₈	—	4	1	no	—
421	C₈	—	268	1	no	—
422	C₈	—	268	1	no	—
423	C₈	—	268	2	no	—
424	C₈	—	134	4	no	—
425	C₂²	—	2	1	no	—
426	C₂²	—	2	1	no	—
427	C₂²	—	2	1	no	—
428	C₂²	—	268	4	no	—
429	C₂²	—	67	2	no	—
430	C₂²	—	134	1	no	—
431	C₂²	—	134	1	no	—
432	C₂²	—	134	1	no	—
433	C₂²	—	134	1	no	—
434	C₂²	—	134	1	no	—
435	C₂²	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
436	C₄	—	2	1	no	—
437	C₄	—	2	1	no	—
438	C₄	—	2	1	no	—
439	C₄	—	67	2	no	—
440	C₄	—	134	1	no	—
441	C₄	—	134	1	no	—
442	C₄	—	134	1	no	—
443	C₄	—	134	1	no	—
444	C₄	—	134	1	no	—
445	C₄	#16863443	1	2	yes	—
446	C₄	#131×D₆₇	1	2	yes	—
447	C₄	#16864192	1	1	yes	characteristic
448	C₄	#16864195	1	1	yes	characteristic
449	C₂	—	4	1	no	—
450	C₂	—	268	1	no	—
451	C₂	—	134	2	no	—
452	C₂	#424×D₆₇	1	1	yes	characteristic
453	C₂	#22568796	1	1	yes	characteristic
454	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×#1574	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂³×D₆₇	C₄⋊C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1574	D₆₇
hypocenter	C₆₇	#17808
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1574

Complements

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

N	H ≅ G/N	type
4. #22559000	C₂	semidirect
5. #22558998	C₂	semidirect
6. #22558997	C₂	semidirect
7. #22558988	C₂	semidirect
8. #22558986	C₂	semidirect
9. #22558960	C₂	semidirect
10. #22558958	C₂	semidirect
11. C₆₇×#1574	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
234. #1574	D₆₇	semidirect
239. C₆₇	#17808	semidirect
270. #335	C₂×D₆₇	semidirect
274. #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
#22568841	2
#22568837	2
#431×D₆₇	2
#22559000	2
#22558998	2
#22558997	2
#22558988	2
#22558986	2
#22558960	2
#22558958	2
C₆₇×#1574	2
#17808	67