Subgroups — #30761552

Order 17152.

694 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523953	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863161	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863155	C₂	1	2	yes	maximal
4	#22559906	C₂	1	2	yes	maximal
5	C₆₇×#1735	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11761612	C₂²	1	2	yes	—
7	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
9	C₂×#14523953	—	2	4	no	—
10	C₂×#14523953	C₄	1	4	yes	complemented
11	#16863161	C₄	1	4	yes	complemented
12	#16863155	—	2	4	no	—
13	#16863151	—	2	4	no	—
14	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#117×C₆₇	C₂²	1	2	yes	—
16	#340×C₆₇	C₄	1	4	yes	complemented
17	C₂³×Dic₆₇	—	2	1	no	—
18	C₂×#11761612	—	2	2	no	—
19	C₂×#11761612	—	2	2	no	—
20	C₂×#11761612	—	2	2	no	—
21	C₂×#11761612	—	2	2	no	—
22	C₂×#11761612	—	2	2	no	—
23	C₂×#11761612	C₂×C₄	1	4	yes	—
24	C₂²×C₆₇⋊C₈	—	2	2	no	—
25	C₂²×C₆₇⋊C₈	—	2	2	no	—
26	C₂²×C₆₇⋊C₈	—	2	2	no	—
27	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
28	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
29	C₂²×C₆₇⋊C₈	Q₈	1	2	yes	—
30	C₂×C₄×Dic₆₇	—	2	2	no	—
31	C₂×C₄×Dic₆₇	—	2	1	no	—
32	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
33	C₂×C₄×Dic₆₇	D₄	1	4	yes	—
34	#14523953	—	2	8	no	—
35	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₆₇	—	2	2	no	—
37	C₂²×C₈×C₆₇	—	2	2	no	—
38	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
41	#117×C₆₇	—	2	2	no	—
42	#117×C₆₇	—	2	2	no	—
43	#117×C₆₇	—	2	2	no	—
44	#117×C₆₇	—	2	2	no	—
45	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
46	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
47	C₂×C₄²×C₆₇	—	2	2	no	—
48	C₂²×Dic₆₇	—	2	2	no	—
49	C₂²×Dic₆₇	—	2	4	no	—
50	C₂²×Dic₆₇	—	4	4	no	—
51	#11761612	—	2	4	no	—
52	#11761612	—	2	4	no	—
53	#11761612	—	4	2	no	—
54	#11761612	—	4	2	no	—
55	C₂×C₆₇⋊C₈	—	2	2	no	—
56	C₂×C₆₇⋊C₈	—	2	2	no	—
57	C₂×C₆₇⋊C₈	—	2	4	no	—
58	C₂×C₆₇⋊C₈	—	2	4	no	—
59	C₂×C₆₇⋊C₈	—	2	4	no	—
60	C₂×C₆₇⋊C₈	—	4	2	no	—
61	C₂×C₆₇⋊C₈	—	4	2	no	—
62	C₂×C₆₇⋊C₈	C₄⋊C₄	1	4	yes	—
63	C₄×Dic₆₇	—	2	2	no	—
64	C₄×Dic₆₇	—	2	4	no	—
65	C₄×Dic₆₇	C₂²⋊C₄	1	4	yes	—
66	C₂⁴×C₆₇	#41	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	—	2	4	no	—
68	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	#41	1	2	yes	—
75	C₂²×C₄×C₆₇	#41	1	2	yes	—
76	C₂²×C₄×C₆₇	#41	1	2	yes	—
77	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
78	C₂×C₈×C₆₇	—	2	2	no	—
79	C₂×C₈×C₆₇	—	2	2	no	—
80	C₂×C₈×C₆₇	—	2	4	no	—
81	C₂×C₈×C₆₇	—	2	4	no	—
82	C₂×C₈×C₆₇	C₄²	1	4	yes	—
83	C₄⋊C₄×C₆₇	—	2	4	no	—
84	C₄⋊C₄×C₆₇	—	2	4	no	—
85	C₂²⋊C₄×C₆₇	—	4	2	no	—
86	C₂²⋊C₄×C₆₇	—	4	2	no	—
87	C₄²×C₆₇	—	4	2	no	—
88	C₄²×C₆₇	—	4	2	no	—
89	C₂³×C₆₇	—	2	4	no	—
90	C₂³×C₆₇	#117	1	1	yes	characteristic
91	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
92	C₂³×C₆₇	C₄×Q₈	1	1	yes	characteristic
93	C₂³×C₆₇	#123	1	2	yes	—
94	C₂³×C₆₇	#126	1	2	yes	—
95	C₂×C₄×C₆₇	—	4	4	no	—
96	C₂×C₄×C₆₇	—	4	8	no	—
97	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
98	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
101	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
102	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
103	C₂×C₄×C₆₇	#117	1	2	yes	—
104	C₂×C₄×C₆₇	#117	1	2	yes	—
105	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
109	C₂×C₄×C₆₇	#121	1	2	yes	—
110	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
111	C₂×C₄×C₆₇	#123	1	2	yes	—
112	C₂×C₄×C₆₇	#124	1	2	yes	—
113	C₂×C₄×C₆₇	#125	1	2	yes	—
114	C₂×C₄×C₆₇	#126	1	2	yes	—
115	C₂×Dic₆₇	—	2	4	no	—
116	C₂×Dic₆₇	—	4	8	no	—
117	C₂×Dic₆₇	—	4	4	no	—
118	C₈×C₆₇	—	2	4	no	—
119	C₆₇⋊C₈	—	2	4	no	—
120	C₆₇⋊C₈	—	4	4	no	—
121	C₂²×C₆₇	—	2	4	no	—
122	C₂²×C₆₇	—	2	4	no	—
123	C₂²×C₆₇	—	2	2	no	—
124	C₂²×C₆₇	—	2	2	no	—
125	C₂²×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
126	C₂²×C₆₇	#381	1	1	yes	characteristic
127	C₂²×C₆₇	#383	1	1	yes	characteristic
128	C₂²×C₆₇	#387	1	2	yes	—
129	C₂²×C₆₇	#388	1	2	yes	—
130	C₂²×C₆₇	#444	1	4	yes	—
131	C₄×C₆₇	—	8	4	no	—
132	C₄×C₆₇	#376	1	1	yes	characteristic
133	C₄×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
134	C₄×C₆₇	#380	1	1	yes	characteristic
135	C₄×C₆₇	#382	1	1	yes	characteristic
136	C₄×C₆₇	#387	1	2	yes	—
137	C₄×C₆₇	#388	1	2	yes	—
138	Dic₆₇	—	4	4	no	—
139	#10819	—	67	1	no	maximal, Sylow
140	C₂×C₆₇	—	2	1	no	—
141	C₂×C₆₇	—	2	1	no	—
142	C₂×C₆₇	—	2	2	no	—
143	C₂×C₆₇	#1434	1	1	yes	characteristic
144	C₂×C₆₇	#1773	1	2	yes	—
145	C₂×C₆₇	#1938	1	4	yes	—
146	C₂²×#97	—	67	1	no	—
147	C₂×#341	—	67	1	no	—
148	#1735	D₆₇	1	1	yes	characteristic
149	C₂×#339	—	67	2	no	—
150	#1729	—	67	2	no	—
151	C₆₇	#10819	1	1	yes	characteristic, complemented, Sylow
152	C₂²×M₁₆	—	67	2	no	—
153	C₂³×C₈	—	67	1	no	—
154	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
155	C₂×#117	C₂×D₆₇	1	2	yes	—
156	C₂²×C₄²	—	67	1	no	—
157	C₂×#97	—	67	4	no	—
158	C₂×#97	—	134	4	no	—
159	#341	—	67	4	no	—
160	#340	Dic₆₇	1	4	yes	complemented
161	#339	—	134	4	no	—
162	#336	—	134	4	no	—
163	C₂³×C₄	—	134	1	no	—
164	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
165	C₂×M₁₆	—	67	4	no	—
166	C₂×M₁₆	—	134	2	no	—
167	C₂×M₁₆	—	134	2	no	—
168	C₂×M₁₆	—	134	2	no	—
169	C₂×M₁₆	—	134	2	no	—
170	C₂×M₁₆	—	134	2	no	—
171	C₂²×C₈	—	2	2	no	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	—	67	2	no	—
174	C₂²×C₈	—	67	2	no	—
175	C₂²×C₈	—	67	2	no	—
176	C₂²×C₈	—	134	2	no	—
177	C₂²×C₈	—	134	2	no	—
178	C₂²×C₈	—	134	2	no	—
179	C₂²×C₈	C₄×D₆₇	1	2	yes	—
180	C₂²×C₈	C₄×D₆₇	1	2	yes	—
181	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
182	#117	—	2	2	no	—
183	#117	—	2	2	no	—
184	#117	—	2	2	no	—
185	#117	—	2	2	no	—
186	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
187	C₂×C₂²⋊C₄	—	2	2	no	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	—	67	2	no	—
190	C₂×C₄²	—	67	4	no	—
191	C₂×C₄²	—	134	2	no	—
192	C₂×C₄²	—	134	1	no	—
193	#97	—	134	8	no	—
194	C₂⁴	#11761627	1	1	yes	characteristic
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	268	4	no	—
197	C₂²×C₄	—	134	2	no	—
198	C₂²×C₄	—	134	4	no	—
199	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
201	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
202	C₂²×C₄	D₄×D₆₇	1	2	yes	—
203	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
204	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
205	C₂²×C₄	#11761627	1	2	yes	—
206	C₂²×C₄	#11761628	1	2	yes	—
207	C₂²×C₄	#11761628	1	2	yes	—
208	C₂²×C₄	#11761629	1	1	yes	characteristic
209	M₁₆	—	268	2	no	—
210	M₁₆	—	268	2	no	—
211	M₁₆	—	134	4	no	—
212	M₁₆	—	134	4	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	268	2	no	—
218	C₂×C₈	—	268	2	no	—
219	C₂×C₈	—	67	4	no	—
220	C₂×C₈	—	134	4	no	—
221	C₂×C₈	—	134	2	no	—
222	C₂×C₈	—	134	2	no	—
223	C₂×C₈	—	134	4	no	—
224	C₂×C₈	—	134	4	no	—
225	C₂×C₈	C₄×Dic₆₇	1	4	yes	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₂²⋊C₄	—	4	2	no	—
229	C₂²⋊C₄	—	4	2	no	—
230	C₄²	—	4	2	no	—
231	C₄²	—	4	2	no	—
232	C₄²	—	67	4	no	—
233	C₄²	—	134	2	no	—
234	C₄²	—	134	4	no	—
235	C₂³	—	2	4	no	—
236	C₂³	#14524001	1	1	yes	characteristic
237	C₂³	D₄×Dic₆₇	1	1	yes	characteristic
238	C₂³	#14524008	1	1	yes	characteristic
239	C₂³	#14524025	1	2	yes	—
240	C₂³	#14524036	1	2	yes	—
241	C₂×C₄	—	4	8	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	268	4	no	—
244	C₂×C₄	—	268	8	no	—
245	C₂×C₄	—	134	4	no	—
246	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
247	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
248	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
249	C₂×C₄	#14523999	1	1	yes	characteristic
250	C₂×C₄	#14524000	1	1	yes	characteristic
251	C₂×C₄	#14524000	1	1	yes	characteristic
252	C₂×C₄	#14524000	1	2	yes	—
253	C₂×C₄	#14524002	1	2	yes	—
254	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	2	yes	—
255	C₂×C₄	#14524005	1	1	yes	characteristic
256	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
257	C₂×C₄	Q₈×Dic₆₇	1	1	yes	characteristic
258	C₂×C₄	#14524016	1	2	yes	—
259	C₂×C₄	#14524021	1	2	yes	—
260	C₂×C₄	#14524028	1	2	yes	—
261	C₂×C₄	#14524029	1	2	yes	—
262	C₂×C₄	#14524033	1	2	yes	—
263	C₂×C₄	#14524038	1	2	yes	—
264	C₈	—	2	4	no	—
265	C₈	—	268	4	no	—
266	C₈	—	134	4	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	2	4	no	—
271	C₂²	C₄⋊C₄×Dic₆₇	1	1	yes	characteristic
272	C₂²	#16863258	1	1	yes	characteristic
273	C₂²	#16863263	1	1	yes	characteristic
274	C₂²	#16863278	1	2	yes	—
275	C₂²	#16863289	1	2	yes	—
276	C₂²	#16863518	1	4	yes	—
277	C₄	—	8	4	no	—
278	C₄	—	268	4	no	—
279	C₄	#16863242	1	1	yes	characteristic
280	C₄	C₂²⋊C₄×Dic₆₇	1	1	yes	characteristic
281	C₄	#16863255	1	1	yes	characteristic
282	C₄	#16863260	1	1	yes	characteristic
283	C₄	#16863282	1	2	yes	—
284	C₄	#16863285	1	2	yes	—
285	C₂	—	2	1	no	—
286	C₂	—	2	1	no	—
287	C₂	—	2	2	no	—
288	C₂	#22558434	1	1	yes	characteristic
289	C₂	#22560095	1	2	yes	—
290	C₂	#22560922	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₆₇×#1735	C₂
socle	C₂³×C₆₇	C₄×D₄
prime radical	C₂⁴×C₆₇	#41
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	#1735	D₆₇
hypocenter	C₆₇	#10819
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #1735

Complements

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

N	H ≅ G/N	type
10. C₂×#14523953	C₄	semidirect
11. #16863161	C₄	semidirect
16. #340×C₆₇	C₄	semidirect
151. C₆₇	#10819	semidirect
160. #340	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14523953	2
C₂×#16863161	2
C₂×#16863155	2
#22559906	2
C₆₇×#1735	2
#10819	67