Subgroups — #30764029

Order 17152.

654 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523979	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863155	C₂	1	2	yes	maximal
3	#22559906	C₂	1	2	yes	maximal
4	C₂×#16863150	C₂	1	1	yes	maximal, characteristic
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×#14523979	—	2	4	no	—
10	C₂×#14523979	—	2	2	no	—
11	#16863155	—	2	4	no	—
12	#16863151	—	2	4	no	—
13	#16863150	—	2	2	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	D₄	1	4	yes	—
24	C₂²×C₆₇⋊C₈	—	2	1	no	—
25	C₂²×C₆₇⋊C₈	—	2	2	no	—
26	C₂²×C₆₇⋊C₈	—	2	2	no	—
27	C₂²×C₆₇⋊C₈	—	2	2	no	—
28	C₂²×C₆₇⋊C₈	—	2	1	no	—
29	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
30	C₂×C₄×Dic₆₇	—	2	2	no	—
31	C₂×C₄×Dic₆₇	—	2	1	no	—
32	C₂×C₄×Dic₆₇	—	2	2	no	—
33	C₂×C₄×Dic₆₇	Q₈	1	2	yes	—
34	#14523979	—	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	D₄	1	2	yes	—
40	C₂²×C₈×C₆₇	Q₈	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	4	no	—
49	C₂²×Dic₆₇	—	2	2	no	—
50	C₂²×Dic₆₇	—	4	2	no	—
51	C₂²×Dic₆₇	—	4	2	no	—
52	#11761612	—	2	4	no	—
53	#11761612	—	2	4	no	—
54	#11761612	—	4	2	no	—
55	#11761612	—	4	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	—	4	2	no	—
60	C₂×C₆₇⋊C₈	—	4	2	no	—
61	C₂×C₆₇⋊C₈	—	4	1	no	—
62	C₂×C₆₇⋊C₈	—	4	1	no	—
63	C₂×C₆₇⋊C₈	—	4	2	no	—
64	C₄×Dic₆₇	—	2	4	no	—
65	C₄×Dic₆₇	—	4	1	no	—
66	C₄×Dic₆₇	—	4	1	no	—
67	C₂⁴×C₆₇	#41	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	—	2	4	no	—
69	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	#41	1	2	yes	—
76	C₂²×C₄×C₆₇	#41	1	2	yes	—
77	C₂²×C₄×C₆₇	#41	1	2	yes	—
78	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
79	C₂×C₈×C₆₇	—	2	4	no	—
80	C₂×C₈×C₆₇	—	2	2	no	—
81	C₂×C₈×C₆₇	—	2	4	no	—
82	C₂×C₈×C₆₇	—	2	2	no	—
83	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
84	C₄⋊C₄×C₆₇	—	2	4	no	—
85	C₄⋊C₄×C₆₇	—	2	4	no	—
86	C₂²⋊C₄×C₆₇	—	4	2	no	—
87	C₂²⋊C₄×C₆₇	—	4	2	no	—
88	C₄²×C₆₇	—	4	2	no	—
89	C₄²×C₆₇	—	4	2	no	—
90	C₂³×C₆₇	—	2	2	no	—
91	C₂³×C₆₇	—	2	2	no	—
92	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
93	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂³×C₆₇	#124	1	2	yes	—
95	C₂³×C₆₇	#125	1	1	yes	characteristic
96	C₂³×C₆₇	#126	1	2	yes	—
97	C₂×C₄×C₆₇	—	4	4	no	—
98	C₂×C₄×C₆₇	—	4	4	no	—
99	C₂×C₄×C₆₇	—	4	4	no	—
100	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₆₇	#117	1	2	yes	—
102	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
104	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
105	C₂×C₄×C₆₇	#121	1	2	yes	—
106	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₆₇	#123	1	2	yes	—
110	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
111	C₂×C₄×C₆₇	#123	1	2	yes	—
112	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
113	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
114	C₂×C₄×C₆₇	#125	1	2	yes	—
115	C₂×C₄×C₆₇	#125	1	2	yes	—
116	C₂×C₄×C₆₇	#126	1	2	yes	—
117	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
118	C₂×Dic₆₇	—	2	4	no	—
119	C₂×Dic₆₇	—	4	8	no	—
120	C₂×Dic₆₇	—	4	2	no	—
121	C₂×Dic₆₇	—	4	2	no	—
122	C₈×C₆₇	—	2	4	no	—
123	C₆₇⋊C₈	—	4	4	no	—
124	C₆₇⋊C₈	—	4	2	no	—
125	C₂²×C₆₇	—	2	4	no	—
126	C₂²×C₆₇	—	2	4	no	—
127	C₂²×C₆₇	—	2	1	no	—
128	C₂²×C₆₇	—	2	1	no	—
129	C₂²×C₆₇	—	2	1	no	—
130	C₂²×C₆₇	—	2	1	no	—
131	C₂²×C₆₇	#388	1	2	yes	—
132	C₂²×C₆₇	#389	1	1	yes	characteristic
133	C₂²×C₆₇	#393	1	1	yes	characteristic
134	C₂²×C₆₇	#400	1	1	yes	characteristic
135	C₂²×C₆₇	#400	1	2	yes	—
136	C₂²×C₆₇	#443	1	2	yes	—
137	C₂²×C₆₇	#444	1	2	yes	—
138	C₄×C₆₇	—	8	4	no	—
139	C₄×C₆₇	#382	1	1	yes	characteristic
140	C₄×C₆₇	#385	1	1	yes	characteristic
141	C₄×C₆₇	#387	1	2	yes	—
142	C₄×C₆₇	#394	1	1	yes	characteristic
143	C₄×C₆₇	#397	1	1	yes	characteristic
144	C₄×C₆₇	#398	1	2	yes	—
145	Dic₆₇	—	4	4	no	—
146	#11258	—	67	1	no	maximal, Sylow
147	C₂×C₆₇	—	2	1	no	—
148	C₂×C₆₇	—	2	1	no	—
149	C₂×C₆₇	—	2	2	no	—
150	C₂×C₆₇	#1442	1	1	yes	characteristic
151	C₂×C₆₇	#1929	1	4	yes	—
152	C₂×C₆₇	#1945	1	1	yes	characteristic
153	C₂×C₆₇	#1946	1	1	yes	characteristic
154	C₂²×#108	—	67	1	no	—
155	#1735	D₆₇	1	1	yes	characteristic
156	C₂×#339	—	67	2	no	—
157	#1729	—	67	2	no	—
158	C₂×#336	—	67	1	no	—
159	C₆₇	#11258	1	1	yes	characteristic, complemented, Sylow
160	C₂²×M₁₆	—	67	2	no	—
161	C₂³×C₈	—	67	1	no	—
162	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
163	C₂×#117	C₂×D₆₇	1	2	yes	—
164	C₂²×C₄²	—	67	1	no	—
165	C₂×#108	—	134	4	no	—
166	C₂×#108	—	134	2	no	—
167	#340	Dic₆₇	1	4	yes	complemented
168	#339	—	134	4	no	—
169	#336	—	134	4	no	—
170	#336	—	134	2	no	—
171	C₂³×C₄	—	134	1	no	—
172	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
173	C₂×M₁₆	—	67	4	no	—
174	C₂×M₁₆	—	134	2	no	—
175	C₂×M₁₆	—	134	2	no	—
176	C₂×M₁₆	—	134	2	no	—
177	C₂×M₁₆	—	134	2	no	—
178	C₂×M₁₆	—	134	2	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	67	2	no	—
182	C₂²×C₈	—	134	2	no	—
183	C₂²×C₈	—	134	1	no	—
184	C₂²×C₈	—	134	2	no	—
185	C₂²×C₈	—	134	2	no	—
186	C₂²×C₈	—	134	1	no	—
187	C₂²×C₈	Dic₁₃₄	1	2	yes	—
188	C₂²×C₈	D₂₆₈	1	2	yes	—
189	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
190	#117	—	2	2	no	—
191	#117	—	2	2	no	—
192	#117	—	2	2	no	—
193	#117	—	2	2	no	—
194	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
195	C₂×C₂²⋊C₄	—	2	2	no	—
196	C₂×C₄²	—	2	2	no	—
197	C₂×C₄²	—	67	2	no	—
198	C₂×C₄²	—	134	1	no	—
199	C₂×C₄²	—	134	2	no	—
200	C₂×C₄²	—	134	2	no	—
201	#108	—	134	8	no	—
202	C₂⁴	#11761627	1	1	yes	characteristic
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	268	2	no	—
205	C₂²×C₄	—	268	2	no	—
206	C₂²×C₄	—	134	4	no	—
207	C₂²×C₄	—	134	2	no	—
208	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
209	C₂²×C₄	D₄×D₆₇	1	2	yes	—
210	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
212	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
213	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
214	C₂²×C₄	#11761627	1	2	yes	—
215	C₂²×C₄	#11761628	1	2	yes	—
216	C₂²×C₄	#11761628	1	2	yes	—
217	C₂²×C₄	#11761629	1	1	yes	characteristic
218	M₁₆	—	268	2	no	—
219	M₁₆	—	268	2	no	—
220	M₁₆	—	134	4	no	—
221	M₁₆	—	134	4	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	268	2	no	—
227	C₂×C₈	—	268	2	no	—
228	C₂×C₈	—	268	1	no	—
229	C₂×C₈	—	268	1	no	—
230	C₂×C₈	—	268	2	no	—
231	C₂×C₈	—	134	4	no	—
232	C₂×C₈	—	134	2	no	—
233	C₂×C₈	—	134	4	no	—
234	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₂²⋊C₄	—	4	2	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₄²	—	4	2	no	—
240	C₄²	—	4	2	no	—
241	C₄²	—	268	1	no	—
242	C₄²	—	268	1	no	—
243	C₄²	—	134	4	no	—
244	C₂³	—	2	2	no	—
245	C₂³	—	2	2	no	—
246	C₂³	D₄×Dic₆₇	1	1	yes	characteristic
247	C₂³	#14524023	1	1	yes	characteristic
248	C₂³	#14524029	1	2	yes	—
249	C₂³	#14524034	1	1	yes	characteristic
250	C₂³	#14524036	1	2	yes	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	268	8	no	—
255	C₂×C₄	—	268	2	no	—
256	C₂×C₄	—	268	2	no	—
257	C₂×C₄	—	134	4	no	—
258	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
259	C₂×C₄	#14524002	1	2	yes	—
260	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
261	C₂×C₄	Q₈×Dic₆₇	1	1	yes	characteristic
262	C₂×C₄	#14524013	1	1	yes	characteristic
263	C₂×C₄	#14524016	1	2	yes	—
264	C₂×C₄	#14524018	1	1	yes	characteristic
265	C₂×C₄	#14524020	1	2	yes	—
266	C₂×C₄	#14524021	1	2	yes	—
267	C₂×C₄	#14524025	1	2	yes	—
268	C₂×C₄	#14524027	1	1	yes	characteristic
269	C₂×C₄	#14524028	1	2	yes	—
270	C₂×C₄	#14524031	1	1	yes	characteristic
271	C₂×C₄	#14524031	1	1	yes	characteristic
272	C₂×C₄	#14524033	1	2	yes	—
273	C₂×C₄	#14524035	1	2	yes	—
274	C₂×C₄	#14524038	1	2	yes	—
275	C₂×C₄	#14524042	1	1	yes	characteristic
276	C₈	—	2	4	no	—
277	C₈	—	268	4	no	—
278	C₈	—	268	2	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	#16863289	1	2	yes	—
286	C₂²	#16863295	1	1	yes	characteristic
287	C₂²	#16863309	1	1	yes	characteristic
288	C₂²	#16863336	1	1	yes	characteristic
289	C₂²	#16863338	1	2	yes	—
290	C₂²	#16863513	1	2	yes	—
291	C₂²	#16863518	1	2	yes	—
292	C₄	—	8	4	no	—
293	C₄	—	268	4	no	—
294	C₄	#16863262	1	1	yes	characteristic
295	C₄	#16863271	1	1	yes	characteristic
296	C₄	#16863282	1	2	yes	—
297	C₄	#16863314	1	1	yes	characteristic
298	C₄	#16863323	1	1	yes	characteristic
299	C₄	#16863330	1	2	yes	—
300	C₂	—	2	1	no	—
301	C₂	—	2	1	no	—
302	C₂	—	2	2	no	—
303	C₂	#22558466	1	1	yes	characteristic
304	C₂	#22560859	1	4	yes	—
305	C₂	#22560958	1	1	yes	characteristic
306	C₂	#22560963	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₆₇
commutator	C₂²×C₄×C₆₇	C₂²×C₄
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	#11258
LCS	G ▷ C₂²×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
16. #340×C₆₇	C₄	semidirect
159. C₆₇	#11258	semidirect
167. #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×#14523979	2
C₂×#16863155	2
#22559906	2
C₂×#16863150	2
C₆₇×#1735	2
#11258	67