Subgroups — #30772318

Order 17152.

384 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863812	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863217	C₂	1	1	yes	maximal, characteristic, complemented
3	#22561967	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16863179	C₂	1	1	yes	maximal, characteristic
5	#22560046	C₂	1	1	yes	maximal, characteristic
6	#22558297	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14524049	C₂²	1	1	yes	characteristic
9	#16863812	—	2	2	no	—
10	#16863812	—	2	2	no	—
11	C₂×#14523983	C₂²	1	1	yes	characteristic
12	C₂×#14523983	C₂²	1	1	yes	characteristic
13	C₂×#14523982	—	2	1	no	—
14	C₂×C₁₆×D₆₇	—	2	1	no	—
15	C₂×#14523971	C₄	1	1	yes	characteristic
16	#16863393	C₄	1	1	yes	characteristic
17	#16863351	C₂²	1	1	yes	characteristic
18	#16863217	—	2	2	no	—
19	Dic₆₇⋊C₁₆	—	2	1	no	—
20	Dic₆₇⋊C₁₆	—	2	1	no	—
21	#16863179	—	2	2	no	—
22	#16863161	C₄	1	1	yes	characteristic
23	#16863156	C₄	1	1	yes	characteristic
24	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
25	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#11761628	C₂×C₄	1	1	yes	characteristic
28	#14524049	—	2	1	no	—
29	#14524049	—	2	1	no	—
30	#14524049	D₄	1	2	yes	—
31	#14524049	Q₈	1	2	yes	—
32	C₂×#11761612	C₂×C₄	1	1	yes	characteristic
33	C₂×#11761612	C₂×C₄	1	1	yes	characteristic
34	C₂×#11761611	—	2	1	no	—
35	C₂×C₈×D₆₇	—	2	1	no	—
36	#14524002	C₂×C₄	1	1	yes	characteristic
37	#14523983	—	2	1	no	—
38	#14523983	—	2	1	no	—
39	#14523983	—	2	1	no	—
40	#14523983	D₄	1	2	yes	—
41	#14523982	—	2	4	no	—
42	C₂×C₆₇⋊C₁₆	—	2	1	no	—
43	C₂×C₆₇⋊C₁₆	—	2	1	no	—
44	C₁₆×D₆₇	—	2	4	no	—
45	#14523971	—	2	2	no	—
46	#14523968	C₈	1	1	yes	characteristic
47	#14523968	C₈	1	1	yes	characteristic
48	#14523965	C₈	1	1	yes	characteristic
49	#14523965	C₈	1	1	yes	characteristic
50	#14523953	—	2	1	no	—
51	C₈×Dic₆₇	—	2	1	no	—
52	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₂×C₁₆×C₆₇	—	2	1	no	—
56	C₂×C₁₆×C₆₇	—	2	1	no	—
57	C₂×C₁₆×C₆₇	—	2	1	no	—
58	C₂×C₁₆×C₆₇	—	2	1	no	—
59	C₂×C₁₆×C₆₇	—	2	1	no	—
60	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
61	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
62	C₄×C₈×C₆₇	—	2	1	no	—
63	C₄×C₈×C₆₇	—	2	1	no	—
64	#11761628	—	2	1	no	—
65	#11761628	—	2	1	no	—
66	#11761628	C₄⋊C₄	1	2	yes	—
67	#11761628	M₁₆	1	2	yes	—
68	C₂×Dic₁₃₄	C₂×C₈	1	1	yes	characteristic
69	C₂×D₂₆₈	C₂×C₈	1	1	yes	characteristic
70	C₂×C₂²⋊D₆₇	—	2	1	no	—
71	C₂×C₄×D₆₇	—	2	1	no	—
72	#11761612	—	2	1	no	—
73	#11761612	—	2	1	no	—
74	#11761612	—	2	1	no	—
75	#11761612	C₄⋊C₄	1	2	yes	—
76	#11761611	—	2	2	no	—
77	#11761611	—	2	2	no	—
78	C₂×C₆₇⋊C₈	—	2	1	no	—
79	C₂×C₆₇⋊C₈	—	2	1	no	—
80	C₈×D₆₇	—	2	2	no	—
81	C₈×D₆₇	—	2	2	no	—
82	C₄⋊Dic₆₇	C₂×C₈	1	1	yes	characteristic
83	C₄⋊Dic₆₇	C₂×C₈	1	1	yes	characteristic
84	C₂²⋊Dic₆₇	—	2	1	no	—
85	C₄×Dic₆₇	—	2	1	no	—
86	C₆₇⋊C₁₆	—	2	2	no	—
87	C₆₇⋊C₁₆	—	4	1	no	—
88	C₂²×C₄×C₆₇	—	2	1	no	—
89	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₆₇	—	2	1	no	—
91	C₂×C₈×C₆₇	—	2	1	no	—
92	C₂×C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
93	C₂×C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
94	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
98	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
99	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
100	C₄²×C₆₇	—	2	1	no	—
101	C₄²×C₆₇	—	2	1	no	—
102	C₁₆×C₆₇	—	2	1	no	—
103	C₁₆×C₆₇	—	2	1	no	—
104	C₁₆×C₆₇	—	4	1	no	—
105	C₂³×C₆₇	#131	1	1	yes	characteristic
106	C₂²×D₆₇	—	2	1	no	—
107	C₂×C₄×C₆₇	—	2	2	no	—
108	C₂×C₄×C₆₇	—	4	1	no	—
109	C₂×C₄×C₆₇	—	4	1	no	—
110	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
115	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
116	C₂²⋊D₆₇	—	2	2	no	—
117	C₂²⋊D₆₇	—	2	2	no	—
118	C₂×Dic₆₇	—	2	1	no	—
119	C₂×Dic₆₇	—	2	1	no	—
120	C₂×Dic₆₇	—	2	1	no	—
121	D₂₆₈	—	2	1	no	—
122	D₂₆₈	C₄⋊C₈	1	2	yes	—
123	C₄×D₆₇	—	2	2	no	—
124	C₄×D₆₇	—	2	2	no	—
125	Dic₁₃₄	—	2	1	no	—
126	Dic₁₃₄	C₄⋊C₈	1	2	yes	—
127	C₈×C₆₇	—	2	1	no	—
128	C₈×C₆₇	—	2	1	no	—
129	C₈×C₆₇	#120	1	1	yes	characteristic
130	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
131	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
132	C₈×C₆₇	#123	1	1	yes	characteristic
133	C₆₇⋊C₈	—	2	2	no	—
134	C₆₇⋊C₈	—	4	1	no	—
135	C₂²×C₆₇	—	2	1	no	—
136	C₂²×C₆₇	—	2	1	no	—
137	C₂²×C₆₇	#424	1	1	yes	characteristic
138	C₂²×C₆₇	C₈×D₄	1	1	yes	characteristic
139	C₂²×C₆₇	#435	1	1	yes	characteristic
140	C₂×D₆₇	—	2	2	no	—
141	C₂×D₆₇	—	4	1	no	—
142	C₂×D₆₇	—	4	1	no	—
143	C₄×C₆₇	—	4	2	no	—
144	C₄×C₆₇	#385	1	1	yes	characteristic
145	C₄×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
146	C₄×C₆₇	C₈×D₄	1	1	yes	characteristic
147	C₄×C₆₇	#435	1	1	yes	characteristic
148	Dic₆₇	—	2	2	no	—
149	Dic₆₇	—	4	1	no	—
150	Dic₆₇	—	4	1	no	—
151	#12917	—	67	1	no	maximal, Sylow
152	C₂×C₆₇	—	2	1	no	—
153	C₂×C₆₇	—	2	1	no	—
154	C₂×C₆₇	#1850	1	1	yes	characteristic
155	C₂×C₆₇	#2170	1	1	yes	characteristic
156	C₂×C₆₇	#2171	1	1	yes	characteristic
157	D₆₇	—	4	2	no	—
158	C₂×#504	—	67	1	no	—
159	C₂×#364	—	67	1	no	—
160	#2151	—	67	1	no	—
161	C₂×#350	—	67	1	no	—
162	#1764	—	67	1	no	—
163	#1379	—	67	1	no	—
164	#1374	D₆₇	1	1	yes	characteristic, complemented
165	C₆₇	#12917	1	1	yes	characteristic, complemented, Sylow
166	C₂×#131	—	67	1	no	—
167	#504	—	134	2	no	—
168	#504	—	134	2	no	—
169	C₂×M₃₂	—	67	1	no	—
170	C₂×M₃₂	—	67	1	no	—
171	C₂×M₃₂	—	134	1	no	—
172	C₂²×C₁₆	—	134	1	no	—
173	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
174	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
175	C₂×#104	—	67	1	no	—
176	#416	—	67	1	no	—
177	#405	—	67	1	no	—
178	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
179	#364	—	134	2	no	—
180	C₄⋊C₁₆	—	134	1	no	—
181	C₄⋊C₁₆	—	134	1	no	—
182	#350	—	134	2	no	—
183	#341	—	67	1	no	—
184	#339	—	67	1	no	—
185	C₂×#41	—	67	1	no	—
186	#131	—	67	2	no	—
187	#131	—	67	2	no	—
188	#131	—	134	1	no	—
189	#131	—	134	1	no	—
190	C₂×M₁₆	—	67	1	no	—
191	C₂×M₁₆	—	67	1	no	—
192	C₂×M₁₆	—	134	1	no	—
193	C₂²×C₈	—	134	1	no	—
194	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
195	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
196	#117	—	67	1	no	—
197	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
198	M₃₂	—	67	2	no	—
199	M₃₂	—	134	4	no	—
200	M₃₂	—	134	1	no	—
201	M₃₂	—	134	1	no	—
202	M₃₂	—	134	1	no	—
203	C₂×C₁₆	—	2	1	no	—
204	C₂×C₁₆	—	2	1	no	—
205	C₂×C₁₆	—	2	1	no	—
206	C₂×C₁₆	—	2	1	no	—
207	C₂×C₁₆	—	2	1	no	—
208	C₂×C₁₆	—	134	4	no	—
209	C₂×C₁₆	—	134	1	no	—
210	C₂×C₁₆	—	134	1	no	—
211	C₂×C₁₆	D₂₆₈	1	1	yes	characteristic
212	C₂×C₁₆	D₂₆₈	1	1	yes	characteristic
213	#104	—	134	2	no	—
214	#103	—	67	1	no	—
215	#103	—	67	1	no	—
216	#102	—	67	1	no	—
217	#102	—	67	1	no	—
218	#97	—	134	1	no	—
219	C₄×C₈	—	2	1	no	—
220	C₄×C₈	—	2	1	no	—
221	C₄×C₈	—	134	1	no	—
222	#41	—	67	2	no	—
223	#41	—	67	2	no	—
224	#41	—	134	1	no	—
225	#41	—	134	1	no	—
226	C₂×Q₈	—	67	1	no	—
227	C₂×D₄	—	67	1	no	—
228	C₂×D₄	—	134	1	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	134	1	no	—
231	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
232	M₁₆	—	67	2	no	—
233	M₁₆	—	134	1	no	—
234	M₁₆	—	134	1	no	—
235	M₁₆	—	134	1	no	—
236	M₁₆	—	134	2	no	—
237	M₁₆	—	134	2	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	134	1	no	—
241	C₂×C₈	—	134	1	no	—
242	C₂×C₈	—	134	2	no	—
243	C₂×C₈	—	134	2	no	—
244	C₂×C₈	C₈×D₆₇	1	1	yes	characteristic
245	C₂×C₈	C₈×D₆₇	1	1	yes	characteristic
246	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
247	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
248	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
249	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
250	C₂×C₈	#11761627	1	1	yes	characteristic
251	C₂×C₈	#11761628	1	1	yes	characteristic
252	C₄⋊C₄	—	67	1	no	—
253	C₄⋊C₄	—	67	1	no	—
254	C₂²⋊C₄	—	134	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	134	1	no	—
258	C₁₆	—	2	1	no	—
259	C₁₆	—	2	1	no	—
260	C₁₆	—	4	1	no	—
261	C₁₆	—	268	1	no	—
262	C₁₆	—	134	2	no	—
263	C₂³	—	134	1	no	—
264	C₂³	#14524049	1	1	yes	characteristic
265	Q₈	—	67	2	no	—
266	Q₈	—	134	1	no	—
267	D₄	—	67	2	no	—
268	D₄	—	134	2	no	—
269	D₄	—	134	2	no	—
270	D₄	—	134	1	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	134	1	no	—
275	C₂×C₄	—	134	1	no	—
276	C₂×C₄	—	134	1	no	—
277	C₂×C₄	—	134	2	no	—
278	C₂×C₄	—	134	2	no	—
279	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
280	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
281	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
282	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
283	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
284	C₂×C₄	#14524051	1	1	yes	characteristic
285	C₈	—	2	1	no	—
286	C₈	—	2	1	no	—
287	C₈	—	268	1	no	—
288	C₈	—	134	2	no	—
289	C₈	#14524013	1	1	yes	characteristic
290	C₈	#14524021	1	1	yes	characteristic
291	C₈	#14524023	1	1	yes	characteristic
292	C₈	#14524025	1	1	yes	characteristic
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	268	1	no	—
296	C₂²	—	268	1	no	—
297	C₂²	—	134	2	no	—
298	C₂²	#16863421	1	1	yes	characteristic
299	C₂²	C₈×D₂₆₈	1	1	yes	characteristic
300	C₂²	#16863470	1	1	yes	characteristic
301	C₄	—	4	2	no	—
302	C₄	—	268	1	no	—
303	C₄	—	268	1	no	—
304	C₄	—	134	2	no	—
305	C₄	#16863270	1	1	yes	characteristic
306	C₄	C₄⋊C₈×D₆₇	1	1	yes	characteristic
307	C₄	(C₂²⋊D₆₇)⋊C₈	1	1	yes	characteristic
308	C₄	#16863469	1	1	yes	characteristic
309	C₂	—	2	1	no	—
310	C₂	—	2	1	no	—
311	C₂	—	268	2	no	—
312	C₂	#22560388	1	1	yes	characteristic
313	C₂	#22562047	1	1	yes	characteristic
314	C₂	#22562052	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1374	C₂
socle	C₂²×C₆₇	C₈×D₄
prime radical	C₂×#11761628	C₂×C₄
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1374	D₆₇
hypocenter	C₆₇	#12917
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ #1374

Complements

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

N	H ≅ G/N	type
2. C₂×#16863217	C₂	semidirect
3. #22561967	C₂	semidirect
6. #22558297	C₂	semidirect
7. C₆₇×#1374	C₂	semidirect
164. #1374	D₆₇	semidirect
165. C₆₇	#12917	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863812	2
C₂×#16863217	2
#22561967	2
C₂×#16863179	2
#22560046	2
#22558297	2
C₆₇×#1374	2
#12917	67