Subgroups — #30763924

Order 17152.

638 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	C₂²×#14523979	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863168	C₂	1	2	yes	maximal
3	#22559926	C₂	1	1	yes	maximal, characteristic
4	#22559918	C₂	1	2	yes	maximal
5	C₂×#424×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11761612	C₂²	1	2	yes	—
7	C₂×#14524002	C₂²	1	2	yes	—
8	C₂×#14523979	—	2	4	no	—
9	C₂×#14523979	—	2	2	no	—
10	#16863168	—	2	4	no	—
11	#16863159	—	2	2	no	—
12	#16863155	—	2	4	no	—
13	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
16	#424×C₆₇	—	2	2	no	—
17	#424×C₆₇	C₄	1	4	yes	complemented
18	C₂×C₄⋊C₈×C₆₇	—	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	—	2	2	no	—
24	C₂×#11761612	D₄	1	4	yes	—
25	C₂²×C₆₇⋊C₈	—	2	2	no	—
26	#14524002	—	2	2	no	—
27	#14524002	—	2	2	no	—
28	#14524002	—	2	2	no	—
29	#14524002	Q₈	1	4	yes	—
30	C₂×C₄⋊Dic₆₇	—	2	2	no	—
31	C₂×C₂²⋊Dic₆₇	—	2	2	no	—
32	C₂×C₄×Dic₆₇	—	2	2	no	—
33	#14523979	—	2	8	no	—
34	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₆₇	—	2	1	no	—
36	C₂×M₁₆×C₆₇	—	2	2	no	—
37	C₂×M₁₆×C₆₇	—	2	2	no	—
38	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₆₇	—	2	2	no	—
40	C₂²×C₈×C₆₇	—	2	2	no	—
41	C₂²×C₈×C₆₇	—	2	1	no	—
42	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₆₇	D₄	1	2	yes	—
44	C₂²×C₈×C₆₇	D₄	1	2	yes	—
45	#117×C₆₇	—	2	1	no	—
46	#117×C₆₇	—	2	1	no	—
47	#117×C₆₇	C₂×C₄	1	2	yes	—
48	#117×C₆₇	D₄	1	2	yes	—
49	C₂×C₄⋊C₄×C₆₇	Q₈	1	2	yes	—
50	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
51	C₂×C₄²×C₆₇	—	2	1	no	—
52	C₄⋊C₈×C₆₇	—	2	8	no	—
53	C₂²×Dic₆₇	—	4	2	no	—
54	#11761612	—	2	4	no	—
55	#11761612	—	2	4	no	—
56	#11761612	—	4	2	no	—
57	#11761612	—	4	2	no	—
58	C₂×C₆₇⋊C₈	—	2	4	no	—
59	C₂×C₆₇⋊C₈	—	4	2	no	—
60	C₂×C₆₇⋊C₈	—	4	2	no	—
61	C₄⋊Dic₆₇	—	2	4	no	—
62	C₄⋊Dic₆₇	—	4	2	no	—
63	C₂²⋊Dic₆₇	—	2	4	no	—
64	C₂²⋊Dic₆₇	—	4	2	no	—
65	C₄×Dic₆₇	—	2	4	no	—
66	C₄×Dic₆₇	—	4	2	no	—
67	C₂⁴×C₆₇	#41	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	—	2	2	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×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
75	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	#41	1	2	yes	—
77	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	#41	1	2	yes	—
79	M₁₆×C₆₇	—	2	4	no	—
80	M₁₆×C₆₇	—	4	2	no	—
81	C₂×C₈×C₆₇	—	2	4	no	—
82	C₂×C₈×C₆₇	—	2	2	no	—
83	C₂×C₈×C₆₇	—	2	4	no	—
84	C₂×C₈×C₆₇	—	2	2	no	—
85	C₂×C₈×C₆₇	—	2	2	no	—
86	C₂×C₈×C₆₇	—	4	1	no	—
87	C₂×C₈×C₆₇	—	4	1	no	—
88	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
89	C₄⋊C₄×C₆₇	—	2	2	no	—
90	C₄⋊C₄×C₆₇	C₄⋊C₄	1	4	yes	—
91	C₂²⋊C₄×C₆₇	—	2	2	no	—
92	C₂²⋊C₄×C₆₇	—	2	2	no	—
93	C₄²×C₆₇	—	2	2	no	—
94	C₄²×C₆₇	—	2	2	no	—
95	C₂³×C₆₇	—	4	2	no	—
96	C₂³×C₆₇	#117	1	1	yes	characteristic
97	C₂³×C₆₇	#121	1	1	yes	characteristic
98	C₂³×C₆₇	C₂²⋊Q₈	1	2	yes	—
99	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂³×C₆₇	#126	1	2	yes	—
101	C₂×C₄×C₆₇	—	2	4	no	—
102	C₂×C₄×C₆₇	—	4	4	no	—
103	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
107	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
108	C₂×C₄×C₆₇	#121	1	2	yes	—
109	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
110	C₂×C₄×C₆₇	#121	1	2	yes	—
111	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
115	C₂×C₄×C₆₇	#123	1	2	yes	—
116	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
117	C₂×C₄×C₆₇	#125	1	2	yes	—
118	C₂×C₄×C₆₇	#125	1	2	yes	—
119	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
121	C₂×Dic₆₇	—	4	4	no	—
122	C₂×Dic₆₇	—	8	4	no	—
123	C₈×C₆₇	—	2	4	no	—
124	C₈×C₆₇	—	4	2	no	—
125	C₆₇⋊C₈	—	4	4	no	—
126	C₂²×C₆₇	—	2	2	no	—
127	C₂²×C₆₇	—	2	2	no	—
128	C₂²×C₆₇	—	2	2	no	—
129	C₂²×C₆₇	—	4	2	no	—
130	C₂²×C₆₇	—	4	2	no	—
131	C₂²×C₆₇	#380	1	1	yes	characteristic
132	C₂²×C₆₇	#387	1	2	yes	—
133	C₂²×C₆₇	#395	1	1	yes	characteristic
134	C₂²×C₆₇	#396	1	2	yes	—
135	C₂²×C₆₇	#400	1	1	yes	characteristic
136	C₄×C₆₇	—	4	4	no	—
137	C₄×C₆₇	#386	1	1	yes	characteristic
138	C₄×C₆₇	#389	1	2	yes	—
139	C₄×C₆₇	#391	1	1	yes	characteristic
140	C₄×C₆₇	#394	1	1	yes	characteristic
141	C₄×C₆₇	#398	1	1	yes	characteristic
142	C₄×C₆₇	#398	1	2	yes	—
143	Dic₆₇	—	8	4	no	—
144	#11240	—	67	1	no	maximal, Sylow
145	C₂×C₆₇	—	2	2	no	—
146	C₂×C₆₇	—	2	2	no	—
147	C₂×C₆₇	—	2	1	no	—
148	C₂×C₆₇	—	2	1	no	—
149	C₂×C₆₇	#1441	1	1	yes	characteristic
150	C₂×C₆₇	#1814	1	2	yes	—
151	C₂²×#108	—	67	1	no	—
152	C₂×#424	D₆₇	1	1	yes	characteristic
153	C₂×#344	—	67	2	no	—
154	#1735	—	67	1	no	—
155	#1733	—	67	2	no	—
156	C₆₇	#11240	1	1	yes	characteristic, complemented, Sylow
157	C₂²×M₁₆	—	67	2	no	—
158	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
159	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
160	C₂×#117	—	67	2	no	—
161	C₂×#117	C₂×D₆₇	1	1	yes	characteristic
162	C₂×#108	—	134	4	no	—
163	C₂×#108	—	134	2	no	—
164	#424	—	2	2	no	—
165	#424	Dic₆₇	1	4	yes	complemented
166	C₂×C₄⋊C₈	—	2	2	no	—
167	#344	—	134	4	no	—
168	#340	—	134	2	no	—
169	#339	—	134	4	no	—
170	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
171	C₂×M₁₆	—	2	1	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	67	4	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	C₂×M₁₆	—	134	2	no	—
180	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	2	1	no	—
184	C₂²×C₈	—	134	2	no	—
185	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
186	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
187	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
188	#117	—	2	1	no	—
189	#117	—	2	1	no	—
190	#117	—	67	4	no	—
191	#117	—	134	2	no	—
192	#117	—	134	2	no	—
193	#117	—	134	2	no	—
194	#117	D₂₆₈	1	2	yes	—
195	#117	C₂×Dic₆₇	1	2	yes	—
196	C₂×C₄⋊C₄	—	134	2	no	—
197	C₂×C₄⋊C₄	Dic₁₃₄	1	2	yes	—
198	C₂×C₂²⋊C₄	—	2	1	no	—
199	C₂×C₂²⋊C₄	—	134	2	no	—
200	C₂×C₄²	—	2	1	no	—
201	C₂×C₄²	—	134	2	no	—
202	#108	—	134	8	no	—
203	C₄⋊C₈	—	2	8	no	—
204	C₂⁴	#11761628	1	1	yes	characteristic
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	268	2	no	—
207	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
208	C₂²×C₄	D₄×D₆₇	1	2	yes	—
209	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
212	C₂²×C₄	Q₈×D₆₇	1	2	yes	—
213	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
214	C₂²×C₄	#11761627	1	2	yes	—
215	C₂²×C₄	#11761628	1	1	yes	characteristic
216	C₂²×C₄	#11761629	1	2	yes	—
217	M₁₆	—	2	4	no	—
218	M₁₆	—	4	2	no	—
219	M₁₆	—	268	2	no	—
220	M₁₆	—	268	2	no	—
221	M₁₆	—	134	4	no	—
222	M₁₆	—	134	4	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	268	2	no	—
231	C₂×C₈	—	268	2	no	—
232	C₂×C₈	—	134	4	no	—
233	C₂×C₈	C₂²⋊Dic₆₇	1	4	yes	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	268	2	no	—
236	C₄⋊C₄	—	134	4	no	—
237	C₄⋊C₄	C₄⋊Dic₆₇	1	4	yes	—
238	C₂²⋊C₄	—	2	2	no	—
239	C₂²⋊C₄	—	2	2	no	—
240	C₂²⋊C₄	—	268	2	no	—
241	C₂²⋊C₄	—	134	4	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	268	2	no	—
245	C₄²	—	134	4	no	—
246	C₂³	—	4	2	no	—
247	C₂³	#14524002	1	1	yes	characteristic
248	C₂³	#14524017	1	1	yes	characteristic
249	C₂³	#14524021	1	2	yes	—
250	C₂³	#14524022	1	1	yes	characteristic
251	C₂³	#14524038	1	2	yes	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	268	4	no	—
255	C₂×C₄	—	536	4	no	—
256	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
257	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
258	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
259	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
260	C₂×C₄	#14524011	1	1	yes	characteristic
261	C₂×C₄	#14524015	1	2	yes	—
262	C₂×C₄	#14524017	1	1	yes	characteristic
263	C₂×C₄	#14524018	1	2	yes	—
264	C₂×C₄	#14524019	1	2	yes	—
265	C₂×C₄	#14524022	1	1	yes	characteristic
266	C₂×C₄	#14524023	1	2	yes	—
267	C₂×C₄	#14524024	1	1	yes	characteristic
268	C₂×C₄	#14524028	1	2	yes	—
269	C₂×C₄	#14524032	1	1	yes	characteristic
270	C₂×C₄	#14524033	1	2	yes	—
271	C₂×C₄	#14524034	1	2	yes	—
272	C₂×C₄	#14524041	1	1	yes	characteristic
273	C₂×C₄	#14524042	1	2	yes	—
274	C₈	—	2	4	no	—
275	C₈	—	4	2	no	—
276	C₈	—	268	4	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	#16863256	1	1	yes	characteristic
283	C₂²	#16863282	1	2	yes	—
284	C₂²	#16863316	1	1	yes	characteristic
285	C₂²	#16863318	1	2	yes	—
286	C₂²	#16863335	1	1	yes	characteristic
287	C₄	—	4	4	no	—
288	C₄	—	536	4	no	—
289	C₄	#16863276	1	1	yes	characteristic
290	C₄	#16863295	1	2	yes	—
291	C₄	#16863303	1	1	yes	characteristic
292	C₄	#16863313	1	1	yes	characteristic
293	C₄	#16863326	1	1	yes	characteristic
294	C₄	#16863329	1	2	yes	—
295	C₂	—	2	2	no	—
296	C₂	—	2	2	no	—
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	#22558462	1	1	yes	characteristic
300	C₂	#22560263	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14524017
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₂×#424×C₆₇	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂⁴×C₆₇	#41
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	C₂×#424	D₆₇
hypocenter	C₆₇	#11240
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#424

Complements

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

N	H ≅ G/N	type
17. #424×C₆₇	C₄	semidirect
156. C₆₇	#11240	semidirect
165. #424	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×#16863168	2
#22559926	2
#22559918	2
C₂×#424×C₆₇	2
#11240	67