Subgroups — #30824902

Order 17152.

668 conjugacy classes of subgroups.

The subgroup lattice has 444 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	#431×D₆₇	C₂	1	1	yes	maximal, characteristic
3	#431×D₆₇	C₂	1	1	yes	maximal, characteristic
4	#22558990	C₂	1	1	yes	maximal, characteristic, complemented
5	#22558989	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558987	C₂	1	1	yes	maximal, characteristic, complemented
7	#22558982	C₂	1	2	yes	maximal, complemented
8	#22558978	C₂	1	2	yes	maximal, complemented
9	#22558956	C₂	1	2	yes	maximal, complemented
10	#22558954	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1572	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄²×D₆₇	C₂²	1	1	yes	characteristic
13	#16863453	C₂²	1	1	yes	characteristic
14	#16863451	—	2	1	no	—
15	#16863449	C₂²	1	1	yes	characteristic
16	#16863449	C₂²	1	1	yes	characteristic
17	#16863448	C₄	1	2	yes	—
18	#16863448	C₂²	1	2	yes	—
19	#16863448	C₂²	1	2	yes	—
20	#16863447	—	2	1	no	—
21	#16863447	C₂²	1	1	yes	characteristic
22	#16863447	C₂²	1	1	yes	characteristic
23	#16863446	—	2	1	no	—
24	#16863446	—	2	1	no	—
25	#16863446	C₄	1	2	yes	—
26	#16863418	C₂²	1	2	yes	—
27	C₄⋊C₈×D₆₇	C₄	1	2	yes	—
28	C₄⋊C₈×D₆₇	C₂²	1	2	yes	—
29	#16863358	—	2	1	no	—
30	C₂²⋊C₈×D₆₇	—	2	1	no	—
31	C₂²⋊C₈×D₆₇	C₄	1	2	yes	—
32	#16863348	—	2	1	no	—
33	C₄×C₈×D₆₇	—	2	1	no	—
34	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
35	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
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	#431×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	2	no	—
51	C₂×C₈×D₆₇	—	2	2	no	—
52	C₂×C₈×D₆₇	C₂×C₄	1	2	yes	—
53	C₂×C₈×D₆₇	C₂×C₄	1	2	yes	—
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)	C₂×C₄	1	2	yes	—
66	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
67	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
68	Dic₆₇⋊C₈	—	2	2	no	—
69	Dic₆₇⋊C₈	—	2	2	no	—
70	Dic₆₇⋊C₈	C₂×C₄	1	2	yes	—
71	Dic₆₇⋊C₈	C₂×C₄	1	2	yes	—
72	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
73	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
74	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
75	(C₂×D₆₇)⋊C₈	—	2	2	no	—
76	(C₂×D₆₇)⋊C₈	—	2	2	no	—
77	(C₂×D₆₇)⋊C₈	C₂×C₄	1	2	yes	—
78	(C₂×D₆₇)⋊C₈	C₂×C₄	1	2	yes	—
79	#14523954	—	2	1	no	—
80	C₄×C₆₇⋊C₈	—	2	1	no	—
81	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
82	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
83	C₈×Dic₆₇	—	2	2	no	—
84	C₈×Dic₆₇	—	2	2	no	—
85	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
86	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
87	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
88	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
89	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
90	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
91	C₂²⋊C₈×C₆₇	—	2	1	no	—
92	C₂²⋊C₈×C₆₇	—	2	1	no	—
93	C₂²⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
94	#97×C₆₇	—	2	1	no	—
95	C₄×C₈×C₆₇	—	2	1	no	—
96	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
97	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
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	—	2	4	no	—
104	C₂×C₄×D₆₇	—	4	2	no	—
105	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
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	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×D₆₇	C₂×Q₈	1	1	yes	characteristic
116	C₂×C₄×D₆₇	C₂×Q₈	1	1	yes	characteristic
117	#11761611	—	4	4	no	—
118	C₂×C₆₇⋊C₈	—	2	2	no	—
119	C₂×C₆₇⋊C₈	—	2	2	no	—
120	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
121	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
122	C₈×D₆₇	—	2	4	no	—
123	C₈×D₆₇	—	2	4	no	—
124	C₈×D₆₇	—	4	4	no	—
125	C₄×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
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	C₂²×C₄	1	1	yes	characteristic
132	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
133	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
134	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
135	C₄×Dic₆₇	C₂×Q₈	1	1	yes	characteristic
136	C₄×Dic₆₇	C₂×Q₈	1	1	yes	characteristic
137	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
138	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
140	C₂×C₈×C₆₇	—	2	2	no	—
141	C₂×C₈×C₆₇	—	2	2	no	—
142	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
143	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
144	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
145	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
146	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
147	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
148	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂²×D₆₇	—	2	2	no	—
150	C₂²×D₆₇	—	2	2	no	—
151	C₂²×D₆₇	—	4	1	no	—
152	C₂²×D₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₂²×D₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₂×C₄×C₆₇	—	2	2	no	—
155	C₂×C₄×C₆₇	—	2	2	no	—
156	C₂×C₄×C₆₇	—	4	1	no	—
157	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
158	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
159	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
161	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
163	C₂×Dic₆₇	—	2	2	no	—
164	C₂×Dic₆₇	—	2	2	no	—
165	C₂×Dic₆₇	—	2	2	no	—
166	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
167	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
168	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
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	C₄×D₆₇	—	2	2	no	—
173	C₄×D₆₇	—	2	2	no	—
174	C₄×D₆₇	—	2	2	no	—
175	C₄×D₆₇	—	2	2	no	—
176	C₄×D₆₇	—	2	2	no	—
177	C₄×D₆₇	—	2	2	no	—
178	C₄×D₆₇	—	2	2	no	—
179	C₄×D₆₇	—	2	2	no	—
180	C₄×D₆₇	—	4	1	no	—
181	C₄×D₆₇	—	4	1	no	—
182	C₄×D₆₇	#131	1	4	yes	—
183	C₄×D₆₇	#135	1	4	yes	—
184	C₈×C₆₇	—	2	2	no	—
185	C₈×C₆₇	—	2	2	no	—
186	C₈×C₆₇	—	4	2	no	—
187	C₈×C₆₇	—	4	2	no	—
188	C₆₇⋊C₈	—	2	2	no	—
189	C₆₇⋊C₈	—	2	2	no	—
190	C₆₇⋊C₈	—	4	2	no	—
191	C₆₇⋊C₈	—	4	2	no	—
192	C₂²×C₆₇	—	2	1	no	—
193	C₂²×C₆₇	—	2	1	no	—
194	C₂²×C₆₇	—	2	1	no	—
195	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
196	C₂×D₆₇	—	2	1	no	—
197	C₂×D₆₇	—	2	1	no	—
198	C₂×D₆₇	—	2	1	no	—
199	C₂×D₆₇	—	2	1	no	—
200	C₂×D₆₇	—	4	4	no	—
201	C₂×D₆₇	#424	1	2	yes	—
202	C₂×D₆₇	#427	1	2	yes	—
203	C₄×C₆₇	—	2	1	no	—
204	C₄×C₆₇	—	2	1	no	—
205	C₄×C₆₇	—	2	1	no	—
206	C₄×C₆₇	—	2	1	no	—
207	C₄×C₆₇	C₂×#131	1	2	yes	—
208	C₄×C₆₇	C₂×#135	1	2	yes	—
209	Dic₆₇	—	2	1	no	—
210	Dic₆₇	—	2	1	no	—
211	Dic₆₇	—	2	1	no	—
212	Dic₆₇	#424	1	2	yes	—
213	Dic₆₇	#427	1	2	yes	—
214	Dic₆₇	#430	1	2	yes	—
215	#17803	—	67	1	no	maximal, Sylow
216	C₂×C₆₇	—	4	1	no	—
217	C₂×C₆₇	C₂×#424	1	1	yes	characteristic
218	C₂×C₆₇	C₂×#427	1	1	yes	characteristic
219	C₂×C₆₇	C₂×#430	1	1	yes	characteristic
220	D₆₇	—	2	2	no	—
221	D₆₇	—	2	2	no	—
222	#2920	—	67	1	no	—
223	C₂×#431	—	67	1	no	—
224	C₂×#431	—	67	1	no	—
225	#1572	—	67	1	no	—
226	#1572	—	67	1	no	—
227	#1572	—	67	1	no	—
228	#1572	D₆₇	1	1	yes	characteristic, complemented
229	#1571	—	67	2	no	—
230	#1570	—	67	2	no	—
231	#1564	—	67	2	no	—
232	#1564	—	67	2	no	—
233	C₆₇	#17803	1	1	yes	characteristic, complemented, Sylow
234	C₂²×C₄²	—	67	1	no	—
235	#432	—	67	1	no	—
236	#432	—	134	1	no	—
237	#432	C₂×D₆₇	1	1	yes	characteristic
238	#431	—	67	1	no	—
239	#431	—	67	2	no	—
240	#431	—	67	2	no	—
241	#431	—	67	1	no	—
242	#431	—	67	1	no	—
243	#431	—	67	1	no	—
244	#431	—	67	2	no	—
245	#431	—	67	2	no	—
246	#431	—	134	1	no	—
247	#431	—	134	1	no	—
248	#431	—	134	1	no	—
249	#431	C₂×D₆₇	1	1	yes	characteristic
250	#431	C₂×D₆₇	1	1	yes	characteristic
251	#423	—	67	2	no	—
252	C₂×C₄⋊C₈	—	67	2	no	—
253	C₂×C₄⋊C₈	—	67	2	no	—
254	#407	—	134	1	no	—
255	C₂×C₂²⋊C₈	—	67	2	no	—
256	C₂×C₂²⋊C₈	—	134	1	no	—
257	C₄×M₁₆	—	134	1	no	—
258	C₂×C₄×C₈	—	67	1	no	—
259	C₂×C₄×C₈	—	67	1	no	—
260	C₂×C₄×C₈	—	134	1	no	—
261	#335	—	67	2	no	—
262	#335	—	67	2	no	—
263	#335	—	67	2	no	—
264	#335	C₂×D₆₇	1	2	yes	complemented
265	#334	—	67	2	no	—
266	#334	—	67	2	no	—
267	#334	—	67	2	no	—
268	#334	C₂×D₆₇	1	2	yes	complemented
269	C₂³×C₄	—	67	1	no	—
270	C₂³×C₄	—	67	1	no	—
271	C₂³×C₄	—	67	1	no	—
272	C₂×M₁₆	—	134	2	no	—
273	C₂²×C₈	—	67	2	no	—
274	C₂²×C₈	—	67	2	no	—
275	C₂²×C₈	—	134	2	no	—
276	C₂×C₄²	—	67	1	no	—
277	C₂×C₄²	—	67	1	no	—
278	C₂×C₄²	—	67	1	no	—
279	C₂×C₄²	—	67	1	no	—
280	C₂×C₄²	—	67	1	no	—
281	C₂×C₄²	—	67	1	no	—
282	C₂×C₄²	—	67	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	C₂²×D₆₇	1	1	yes	characteristic
288	C₄⋊C₈	—	67	2	no	—
289	C₄⋊C₈	—	67	2	no	—
290	C₄⋊C₈	—	67	2	no	—
291	C₄⋊C₈	—	67	2	no	—
292	C₄⋊C₈	—	67	2	no	—
293	C₄⋊C₈	—	134	2	no	—
294	C₄⋊C₈	—	134	2	no	—
295	C₄⋊C₈	C₄×D₆₇	1	2	yes	—
296	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
297	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
298	C₂²⋊C₈	—	2	1	no	—
299	C₂²⋊C₈	—	2	1	no	—
300	C₂²⋊C₈	—	67	2	no	—
301	C₂²⋊C₈	—	67	2	no	—
302	C₂²⋊C₈	—	67	2	no	—
303	C₂²⋊C₈	—	134	2	no	—
304	C₂²⋊C₈	—	134	2	no	—
305	C₂²⋊C₈	—	134	1	no	—
306	C₂²⋊C₈	—	134	1	no	—
307	C₂²⋊C₈	C₄×D₆₇	1	2	yes	—
308	#97	—	2	1	no	—
309	#97	—	134	1	no	—
310	C₄×C₈	—	2	1	no	—
311	C₄×C₈	—	67	2	no	—
312	C₄×C₈	—	67	2	no	—
313	C₄×C₈	—	67	1	no	—
314	C₄×C₈	—	67	1	no	—
315	C₄×C₈	—	134	2	no	—
316	C₄×C₈	—	134	2	no	—
317	C₄×C₈	—	134	1	no	—
318	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
319	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
320	C₂⁴	—	67	1	no	—
321	C₂²×C₄	—	268	2	no	—
322	C₂²×C₄	—	67	1	no	—
323	C₂²×C₄	—	67	1	no	—
324	C₂²×C₄	—	67	1	no	—
325	C₂²×C₄	—	67	1	no	—
326	C₂²×C₄	—	67	1	no	—
327	C₂²×C₄	—	67	1	no	—
328	C₂²×C₄	—	67	1	no	—
329	C₂²×C₄	—	67	1	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	—	134	4	no	—
338	C₂²×C₄	—	134	4	no	—
339	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
340	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
341	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
342	M₁₆	—	268	4	no	—
343	C₂×C₈	—	2	2	no	—
344	C₂×C₈	—	2	2	no	—
345	C₂×C₈	—	268	4	no	—
346	C₂×C₈	—	67	2	no	—
347	C₂×C₈	—	67	2	no	—
348	C₂×C₈	—	134	4	no	—
349	C₂×C₈	—	134	4	no	—
350	C₂×C₈	—	134	2	no	—
351	C₂×C₈	—	134	2	no	—
352	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
353	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
354	C₄²	—	67	1	no	—
355	C₄²	—	67	1	no	—
356	C₄²	—	67	1	no	—
357	C₄²	—	67	1	no	—
358	C₄²	—	67	1	no	—
359	C₄²	—	67	1	no	—
360	C₄²	—	67	1	no	—
361	C₄²	—	67	1	no	—
362	C₄²	—	67	1	no	—
363	C₄²	—	67	1	no	—
364	C₄²	—	67	1	no	—
365	C₄²	—	67	1	no	—
366	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
367	C₄²	D₄×D₆₇	1	1	yes	characteristic
368	C₄²	Q₈×D₆₇	1	1	yes	characteristic
369	C₄²	C₂³×D₆₇	1	1	yes	characteristic
370	C₂³	—	268	1	no	—
371	C₂³	—	67	1	no	—
372	C₂³	—	67	1	no	—
373	C₂³	—	134	2	no	—
374	C₂³	—	134	2	no	—
375	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	4	1	no	—
379	C₂×C₄	—	268	1	no	—
380	C₂×C₄	—	268	1	no	—
381	C₂×C₄	—	67	1	no	—
382	C₂×C₄	—	67	1	no	—
383	C₂×C₄	—	67	1	no	—
384	C₂×C₄	—	67	4	no	—
385	C₂×C₄	—	67	4	no	—
386	C₂×C₄	—	67	1	no	—
387	C₂×C₄	—	67	1	no	—
388	C₂×C₄	—	67	1	no	—
389	C₂×C₄	—	134	2	no	—
390	C₂×C₄	—	134	2	no	—
391	C₂×C₄	—	134	2	no	—
392	C₂×C₄	—	134	2	no	—
393	C₂×C₄	—	134	2	no	—
394	C₂×C₄	—	134	2	no	—
395	C₂×C₄	—	134	2	no	—
396	C₂×C₄	—	134	2	no	—
397	C₂×C₄	—	134	2	no	—
398	C₂×C₄	—	134	2	no	—
399	C₂×C₄	—	134	2	no	—
400	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
401	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
402	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
403	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
404	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
405	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
406	C₈	—	2	2	no	—
407	C₈	—	2	2	no	—
408	C₈	—	4	2	no	—
409	C₈	—	4	2	no	—
410	C₈	—	268	2	no	—
411	C₈	—	268	2	no	—
412	C₈	—	134	2	no	—
413	C₈	—	134	2	no	—
414	C₂²	—	2	1	no	—
415	C₂²	—	2	1	no	—
416	C₂²	—	2	1	no	—
417	C₂²	—	268	4	no	—
418	C₂²	—	67	2	no	—
419	C₂²	—	67	2	no	—
420	C₂²	—	134	1	no	—
421	C₂²	—	134	1	no	—
422	C₂²	—	134	1	no	—
423	C₂²	—	134	1	no	—
424	C₂²	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
425	C₄	—	2	1	no	—
426	C₄	—	2	1	no	—
427	C₄	—	2	1	no	—
428	C₄	—	2	1	no	—
429	C₄	—	67	2	no	—
430	C₄	—	67	2	no	—
431	C₄	—	67	2	no	—
432	C₄	—	134	1	no	—
433	C₄	—	134	1	no	—
434	C₄	—	134	1	no	—
435	C₄	#131×D₆₇	1	2	yes	—
436	C₄	#135×D₆₇	1	2	yes	—
437	C₂	—	4	1	no	—
438	C₂	—	134	2	no	—
439	C₂	—	134	2	no	—
440	C₂	#424×D₆₇	1	1	yes	characteristic
441	C₂	#427×D₆₇	1	1	yes	characteristic
442	C₂	#430×D₆₇	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×#1572	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂³×D₆₇	C₄⋊C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1572	D₆₇
hypocenter	C₆₇	#17803
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1572

Complements

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

N	H ≅ G/N	type
4. #22558990	C₂	semidirect
5. #22558989	C₂	semidirect
6. #22558987	C₂	semidirect
7. #22558982	C₂	semidirect
8. #22558978	C₂	semidirect
9. #22558956	C₂	semidirect
10. #22558954	C₂	semidirect
11. C₆₇×#1572	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
228. #1572	D₆₇	semidirect
233. C₆₇	#17803	semidirect
264. #335	C₂×D₆₇	semidirect
268. #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
#431×D₆₇	2
#22558990	2
#22558989	2
#22558987	2
#22558982	2
#22558978	2
#22558956	2
#22558954	2
C₆₇×#1572	2
#17803	67