Subgroups — #30755929

Order 17152.

432 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568596	C₂	1	1	yes	maximal, characteristic
2	C₄×#14523954	C₂	1	1	yes	maximal, characteristic
3	#22558091	C₂	1	2	yes	maximal
4	#22558091	C₂	1	2	yes	maximal
5	C₆₇×#1767	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523954	—	2	1	no	—
7	C₂×#14523954	C₂²	1	2	yes	—
8	C₂×C₄×C₆₇⋊C₈	—	2	1	no	—
9	#16863294	C₄	1	1	yes	characteristic
10	#16863294	C₄	1	1	yes	characteristic
11	#16863294	C₄	1	1	yes	characteristic
12	#16863294	C₄	1	1	yes	characteristic
13	#16863280	—	2	1	no	—
14	#16863280	—	2	1	no	—
15	#16863280	—	2	1	no	—
16	#16863280	—	2	1	no	—
17	#16863262	C₂²	1	1	yes	characteristic
18	#16863262	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
20	C₂×#97×C₆₇	C₂²	1	2	yes	—
21	C₄³×C₆₇	C₂²	1	1	yes	characteristic
22	#336×C₆₇	—	2	1	no	—
23	C₂²×C₆₇⋊C₈	—	2	2	no	—
24	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂×Dic₆₇⋊C₄	—	2	1	no	—
29	C₂×Dic₆₇⋊C₄	—	2	1	no	—
30	C₂×Dic₆₇⋊C₄	—	2	1	no	—
31	C₂×Dic₆₇⋊C₄	—	2	1	no	—
32	#14523954	—	2	4	no	—
33	#14523954	—	2	2	no	—
34	#14523954	Q₈	1	4	yes	—
35	C₄×C₆₇⋊C₈	—	2	4	no	—
36	C₆₇⋊#95	—	2	1	no	—
37	C₆₇⋊#95	—	2	1	no	—
38	C₆₇⋊#95	—	2	1	no	—
39	C₆₇⋊#95	—	2	1	no	—
40	C₆₇⋊#95	—	2	1	no	—
41	C₆₇⋊#95	—	2	1	no	—
42	C₆₇⋊#95	—	2	1	no	—
43	C₆₇⋊#95	—	2	1	no	—
44	C₂²×C₈×C₆₇	—	2	2	no	—
45	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
52	C₄⋊C₈×C₆₇	—	2	4	no	—
53	#97×C₆₇	—	2	4	no	—
54	C₂²×Dic₆₇	—	2	1	no	—
55	C₂²×Dic₆₇	—	2	1	no	—
56	C₂²×Dic₆₇	—	2	1	no	—
57	C₂²×Dic₆₇	—	2	1	no	—
58	C₂×C₆₇⋊C₈	—	2	4	no	—
59	C₂×C₆₇⋊C₈	—	2	4	no	—
60	C₂×C₆₇⋊C₈	—	2	4	no	—
61	C₄⋊Dic₆₇	—	4	1	no	—
62	C₄⋊Dic₆₇	—	4	1	no	—
63	C₄⋊Dic₆₇	—	4	1	no	—
64	C₄⋊Dic₆₇	—	4	1	no	—
65	Dic₆₇⋊C₄	—	4	2	no	—
66	Dic₆₇⋊C₄	—	4	2	no	—
67	Dic₆₇⋊C₄	—	4	2	no	—
68	Dic₆₇⋊C₄	—	4	2	no	—
69	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₆₇	—	2	4	no	—
77	C₂×C₈×C₆₇	—	4	2	no	—
78	C₂×C₈×C₆₇	—	4	2	no	—
79	C₄²×C₆₇	—	2	2	no	—
80	C₄²×C₆₇	—	2	2	no	—
81	C₄²×C₆₇	—	2	2	no	—
82	C₄²×C₆₇	—	4	1	no	—
83	C₄²×C₆₇	—	4	1	no	—
84	C₄²×C₆₇	—	4	1	no	—
85	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
86	C₄²×C₆₇	#41	1	2	yes	—
87	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₆₇	—	2	1	no	—
89	C₂×C₄×C₆₇	—	2	1	no	—
90	C₂×C₄×C₆₇	—	2	1	no	—
91	C₂×C₄×C₆₇	—	2	1	no	—
92	C₂×C₄×C₆₇	—	2	1	no	—
93	C₂×C₄×C₆₇	—	2	1	no	—
94	C₂×C₄×C₆₇	—	2	1	no	—
95	C₂×C₄×C₆₇	—	2	1	no	—
96	C₂×C₄×C₆₇	—	2	1	no	—
97	C₂×C₄×C₆₇	—	2	1	no	—
98	C₂×C₄×C₆₇	—	2	1	no	—
99	C₂×C₄×C₆₇	—	2	1	no	—
100	C₂×C₄×C₆₇	—	2	1	no	—
101	C₂×C₄×C₆₇	—	2	1	no	—
102	C₂×C₄×C₆₇	—	2	1	no	—
103	C₂×C₄×C₆₇	#104	1	2	yes	—
104	C₂×C₄×C₆₇	#104	1	2	yes	—
105	C₂×C₄×C₆₇	#104	1	2	yes	—
106	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
107	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
110	C₂×Dic₆₇	—	4	1	no	—
111	C₂×Dic₆₇	—	4	1	no	—
112	C₂×Dic₆₇	—	4	1	no	—
113	C₂×Dic₆₇	—	4	1	no	—
114	C₂×Dic₆₇	—	4	1	no	—
115	C₂×Dic₆₇	—	4	1	no	—
116	C₂×Dic₆₇	—	4	1	no	—
117	C₂×Dic₆₇	—	4	1	no	—
118	C₂×Dic₆₇	—	4	1	no	—
119	C₂×Dic₆₇	—	4	1	no	—
120	C₂×Dic₆₇	—	4	1	no	—
121	C₂×Dic₆₇	—	4	1	no	—
122	C₈×C₆₇	—	4	4	no	—
123	C₆₇⋊C₈	—	2	8	no	—
124	C₂²×C₆₇	#391	1	1	yes	characteristic
125	C₂²×C₆₇	#413	1	1	yes	characteristic
126	C₂²×C₆₇	#413	1	1	yes	characteristic
127	C₂²×C₆₇	C₂×#104	1	1	yes	characteristic
128	C₂²×C₆₇	C₂×#104	1	1	yes	characteristic
129	C₂²×C₆₇	C₂×#104	1	1	yes	characteristic
130	C₂²×C₆₇	#421	1	1	yes	characteristic
131	C₄×C₆₇	—	2	1	no	—
132	C₄×C₆₇	—	2	1	no	—
133	C₄×C₆₇	—	2	2	no	—
134	C₄×C₆₇	—	2	2	no	—
135	C₄×C₆₇	—	2	2	no	—
136	C₄×C₆₇	—	2	2	no	—
137	C₄×C₆₇	—	2	2	no	—
138	C₄×C₆₇	—	4	1	no	—
139	Dic₆₇	—	8	1	no	—
140	Dic₆₇	—	8	1	no	—
141	Dic₆₇	—	8	1	no	—
142	Dic₆₇	—	8	1	no	—
143	#9892	—	67	1	no	maximal, Sylow
144	C₂×C₆₇	#1524	1	1	yes	characteristic
145	C₂×C₆₇	#1524	1	1	yes	characteristic
146	C₂×C₆₇	#1529	1	1	yes	characteristic
147	C₂×C₆₇	#1532	1	1	yes	characteristic
148	C₂×C₆₇	#1986	1	1	yes	characteristic
149	C₂×C₆₇	#1994	1	1	yes	characteristic
150	C₂×C₆₇	#1998	1	1	yes	characteristic
151	#2864	—	67	1	no	—
152	#1767	D₆₇	1	1	yes	characteristic
153	C₄×#97	—	67	1	no	—
154	#1295	—	67	2	no	—
155	#1295	—	67	2	no	—
156	C₆₇	#9892	1	1	yes	characteristic, complemented, Sylow
157	C₂×C₄⋊C₈	—	2	1	no	—
158	C₂×#97	—	67	2	no	—
159	C₂×#97	—	134	1	no	—
160	C₂×#97	C₂×D₆₇	1	2	yes	—
161	C₂×C₄×C₈	—	134	1	no	—
162	#389	—	67	1	no	—
163	#389	—	67	1	no	—
164	#389	—	67	1	no	—
165	#389	—	67	1	no	—
166	#387	—	134	1	no	—
167	#387	—	134	1	no	—
168	#387	—	134	1	no	—
169	#387	—	134	1	no	—
170	#382	—	67	1	no	—
171	#382	—	67	1	no	—
172	C₄³	C₂×D₆₇	1	1	yes	characteristic
173	#336	—	2	1	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	134	2	no	—
176	C₂×C₄⋊C₄	—	67	1	no	—
177	C₂×C₄⋊C₄	—	67	1	no	—
178	C₂×C₄⋊C₄	—	67	1	no	—
179	C₂×C₄⋊C₄	—	67	1	no	—
180	C₂×C₄⋊C₄	—	134	1	no	—
181	C₂×C₄⋊C₄	—	134	1	no	—
182	C₂×C₄⋊C₄	—	134	1	no	—
183	C₂×C₄⋊C₄	—	134	1	no	—
184	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
185	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
186	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
187	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
188	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
189	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
190	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
191	C₄⋊C₈	—	2	4	no	—
192	#97	—	2	4	no	—
193	#97	—	67	4	no	—
194	#97	—	134	4	no	—
195	#97	—	134	2	no	—
196	C₄×C₈	—	134	4	no	—
197	#95	—	134	1	no	—
198	#95	—	134	1	no	—
199	#95	—	134	1	no	—
200	#95	—	134	1	no	—
201	#95	—	134	1	no	—
202	#95	—	134	1	no	—
203	#95	—	134	1	no	—
204	#95	—	134	1	no	—
205	C₂²×C₄	—	134	1	no	—
206	C₂²×C₄	—	134	1	no	—
207	C₂²×C₄	—	134	1	no	—
208	C₂²×C₄	—	134	1	no	—
209	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
210	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
211	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
212	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	—	134	4	no	—
220	C₂×C₈	—	134	4	no	—
221	C₂×C₈	—	134	4	no	—
222	C₄⋊C₄	—	268	1	no	—
223	C₄⋊C₄	—	268	1	no	—
224	C₄⋊C₄	—	268	1	no	—
225	C₄⋊C₄	—	268	1	no	—
226	C₄⋊C₄	—	268	2	no	—
227	C₄⋊C₄	—	268	2	no	—
228	C₄⋊C₄	—	268	2	no	—
229	C₄⋊C₄	—	268	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	4	1	no	—
234	C₄²	—	4	1	no	—
235	C₄²	—	4	1	no	—
236	C₄²	Q₈×D₆₇	1	2	yes	—
237	C₄²	#11761627	1	2	yes	—
238	C₂³	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	268	1	no	—
255	C₂×C₄	—	268	1	no	—
256	C₂×C₄	—	268	1	no	—
257	C₂×C₄	—	268	1	no	—
258	C₂×C₄	—	268	1	no	—
259	C₂×C₄	—	268	1	no	—
260	C₂×C₄	—	268	1	no	—
261	C₂×C₄	—	268	1	no	—
262	C₂×C₄	—	268	1	no	—
263	C₂×C₄	—	268	1	no	—
264	C₂×C₄	—	268	1	no	—
265	C₂×C₄	—	268	1	no	—
266	C₂×C₄	#14523969	1	2	yes	—
267	C₂×C₄	#14523969	1	2	yes	—
268	C₂×C₄	#14523971	1	2	yes	—
269	C₂×C₄	#14524008	1	2	yes	—
270	C₂×C₄	#14524023	1	2	yes	—
271	C₂×C₄	#14524030	1	1	yes	characteristic
272	C₂×C₄	#14524041	1	1	yes	characteristic
273	C₈	—	4	4	no	—
274	C₈	—	134	8	no	—
275	C₂²	#16863301	1	1	yes	characteristic
276	C₂²	#16863378	1	1	yes	characteristic
277	C₂²	#16863378	1	1	yes	characteristic
278	C₂²	C₂×#14523969	1	1	yes	characteristic
279	C₂²	C₂×#14523969	1	1	yes	characteristic
280	C₂²	C₂×#14523971	1	1	yes	characteristic
281	C₂²	#16863414	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	2	2	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	—	4	1	no	—
290	C₄	—	536	1	no	—
291	C₄	—	536	1	no	—
292	C₄	—	536	1	no	—
293	C₄	—	536	1	no	—
294	C₂	#22558795	1	1	yes	characteristic
295	C₂	#22558798	1	1	yes	characteristic
296	C₂	#22558817	1	1	yes	characteristic
297	C₂	#22558830	1	1	yes	characteristic
298	C₂	#22561152	1	1	yes	characteristic
299	C₂	#22561185	1	1	yes	characteristic
300	C₂	#22561201	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₆₇)⋊C₄
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1767	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1767	D₆₇
hypocenter	C₆₇	#9892
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1767

Complements

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

N	H ≅ G/N	type
156. C₆₇	#9892	semidirect

Maximal subgroups

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

subgroup	index
#22568596	2
C₄×#14523954	2
#22558091	2
C₆₇×#1767	2
#9892	67