Subgroups — #30775901

Order 17152.

378 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863224	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16863222	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16863198	C₂	1	1	yes	maximal, characteristic
4	C₂×#16863198	C₂	1	1	yes	maximal, characteristic
5	#22561075	C₂	1	1	yes	maximal, characteristic
6	#22560373	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₅₃₆	C₂²	1	1	yes	characteristic
9	C₂×#14523977	C₂²	1	1	yes	characteristic
10	C₂×#14523977	C₂²	1	1	yes	characteristic
11	C₂×#14523977	C₂²	1	1	yes	characteristic
12	C₂×#14523975	—	2	1	no	—
13	C₂×#14523965	—	2	1	no	—
14	#16863298	—	2	1	no	—
15	#16863262	—	2	1	no	—
16	#16863224	—	2	2	no	—
17	#16863222	—	2	2	no	—
18	#16863198	—	2	2	no	—
19	#16863198	—	2	2	no	—
20	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×D₂₆₈	—	2	1	no	—
24	C₂×D₅₃₆	—	2	1	no	—
25	C₂×D₅₃₆	—	2	1	no	—
26	C₂×D₅₃₆	—	2	1	no	—
27	C₂×D₅₃₆	—	2	1	no	—
28	C₂×D₅₃₆	—	2	1	no	—
29	C₂×D₅₃₆	Q₈	1	2	yes	—
30	C₂×C₄⋊Dic₆₇	—	2	1	no	—
31	C₂×C₄⋊Dic₆₇	—	2	1	no	—
32	C₂×C₄⋊Dic₆₇	—	2	1	no	—
33	C₂×(C₂×D₆₇)⋊C₄	—	4	2	no	—
34	#14523977	—	2	1	no	—
35	#14523977	—	2	1	no	—
36	#14523977	—	2	1	no	—
37	#14523977	—	2	1	no	—
38	#14523977	—	2	1	no	—
39	#14523977	—	2	1	no	—
40	#14523975	—	4	2	no	—
41	#14523965	—	4	2	no	—
42	C₆₇⋊#95	—	4	2	no	—
43	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
46	C₂×C₁₆×C₆₇	—	2	1	no	—
47	C₂×C₁₆×C₆₇	—	2	1	no	—
48	C₂×C₁₆×C₆₇	—	2	1	no	—
49	C₂×C₁₆×C₆₇	—	2	1	no	—
50	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
51	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
52	C₄×C₈×C₆₇	—	2	1	no	—
53	C₄×C₈×C₆₇	—	2	1	no	—
54	C₂³×D₆₇	—	4	1	no	—
55	C₂×D₂₆₈	—	2	2	no	—
56	C₂×D₂₆₈	—	4	1	no	—
57	C₂×D₂₆₈	—	4	1	no	—
58	C₂×D₂₆₈	—	4	1	no	—
59	C₂×D₂₆₈	—	4	1	no	—
60	C₂×D₂₆₈	—	4	1	no	—
61	C₂²×Dic₆₇	—	4	1	no	—
62	C₂²×Dic₆₇	—	4	1	no	—
63	C₂²×Dic₆₇	—	4	1	no	—
64	D₅₃₆	—	2	2	no	—
65	D₅₃₆	—	2	2	no	—
66	D₅₃₆	—	4	1	no	—
67	D₅₃₆	—	4	1	no	—
68	C₄⋊Dic₆₇	—	4	1	no	—
69	C₄⋊Dic₆₇	—	4	1	no	—
70	C₄⋊Dic₆₇	—	4	1	no	—
71	C₄⋊Dic₆₇	—	4	1	no	—
72	C₄⋊Dic₆₇	—	4	1	no	—
73	C₄⋊Dic₆₇	—	4	1	no	—
74	(C₂×D₆₇)⋊C₄	—	8	4	no	—
75	C₂²×C₄×C₆₇	D₈	1	2	yes	—
76	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₆₇	—	2	1	no	—
78	C₂×C₈×C₆₇	—	2	1	no	—
79	C₂×C₈×C₆₇	D₈	1	2	yes	—
80	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
83	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
84	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
85	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
86	C₄²×C₆₇	—	2	1	no	—
87	C₄²×C₆₇	—	2	1	no	—
88	C₁₆×C₆₇	—	2	2	no	—
89	C₁₆×C₆₇	—	2	2	no	—
90	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
91	C₂²×D₆₇	—	4	2	no	—
92	C₂²×D₆₇	—	8	1	no	—
93	C₂²×D₆₇	—	8	1	no	—
94	C₂²×D₆₇	—	8	2	no	—
95	C₂²×D₆₇	—	8	2	no	—
96	C₂×C₄×C₆₇	—	2	2	no	—
97	C₂×C₄×C₆₇	—	2	2	no	—
98	C₂×C₄×C₆₇	—	2	2	no	—
99	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
101	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
102	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
104	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
105	C₂×Dic₆₇	—	8	1	no	—
106	C₂×Dic₆₇	—	8	1	no	—
107	C₂×Dic₆₇	—	8	1	no	—
108	C₂×Dic₆₇	—	8	2	no	—
109	C₂×Dic₆₇	—	8	2	no	—
110	C₂×Dic₆₇	—	8	2	no	—
111	D₂₆₈	—	4	2	no	—
112	D₂₆₈	—	4	2	no	—
113	D₂₆₈	—	8	1	no	—
114	D₂₆₈	—	8	1	no	—
115	C₈×C₆₇	—	2	2	no	—
116	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
117	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
118	C₈×C₆₇	#123	1	1	yes	characteristic
119	C₈×C₆₇	#123	1	1	yes	characteristic
120	C₂²×C₆₇	#474	1	1	yes	characteristic
121	C₂²×C₆₇	#480	1	1	yes	characteristic
122	C₂²×C₆₇	#493	1	1	yes	characteristic
123	C₂²×C₆₇	#508	1	2	yes	—
124	C₂²×C₆₇	#509	1	2	yes	—
125	C₂×D₆₇	—	8	2	no	—
126	C₂×D₆₇	—	8	4	no	—
127	C₂×D₆₇	—	16	2	no	—
128	C₂×D₆₇	—	16	1	no	—
129	C₂×D₆₇	—	16	1	no	—
130	C₄×C₆₇	—	4	2	no	—
131	C₄×C₆₇	#399	1	1	yes	characteristic
132	C₄×C₆₇	#474	1	1	yes	characteristic
133	C₄×C₆₇	#480	1	1	yes	characteristic
134	C₄×C₆₇	#497	1	1	yes	characteristic
135	Dic₆₇	—	16	1	no	—
136	Dic₆₇	—	16	1	no	—
137	Dic₆₇	—	16	1	no	—
138	#13528	—	67	1	no	maximal, Sylow
139	C₂×C₆₇	#2093	1	1	yes	characteristic
140	C₂×C₆₇	#2228	1	2	yes	—
141	C₂×C₆₇	#2234	1	2	yes	—
142	C₂×C₆₇	#2249	1	1	yes	characteristic
143	C₂×C₆₇	#2250	1	1	yes	characteristic
144	D₆₇	—	16	2	no	—
145	C₂×#367	—	67	1	no	—
146	C₂×#366	—	67	1	no	—
147	C₂×#357	—	67	1	no	—
148	C₂×#357	—	67	1	no	—
149	#1970	—	67	1	no	—
150	#1846	—	67	1	no	—
151	#1374	D₆₇	1	1	yes	characteristic, complemented
152	C₆₇	#13528	1	1	yes	characteristic, complemented, Sylow
153	C₂²×D₈	—	67	1	no	—
154	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
155	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
156	C₂×#107	—	67	1	no	—
157	C₂×#107	—	67	1	no	—
158	C₂×#107	—	67	1	no	—
159	C₂×#106	—	134	1	no	—
160	C₂×#102	—	134	1	no	—
161	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
162	#390	—	134	1	no	—
163	#382	—	134	1	no	—
164	#367	—	134	2	no	—
165	#366	—	134	2	no	—
166	#357	—	134	2	no	—
167	#357	—	134	2	no	—
168	C₂²×D₄	—	134	1	no	—
169	C₂×D₈	—	67	2	no	—
170	C₂×D₈	—	134	1	no	—
171	C₂×D₈	—	134	1	no	—
172	C₂×D₈	—	134	1	no	—
173	C₂×D₈	—	134	1	no	—
174	C₂×D₈	—	134	1	no	—
175	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
176	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
177	C₂×C₄⋊C₄	—	134	1	no	—
178	C₂×C₄⋊C₄	—	134	1	no	—
179	C₂×C₄⋊C₄	—	134	1	no	—
180	C₂×C₂²⋊C₄	—	268	2	no	—
181	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
182	C₂×C₁₆	—	2	1	no	—
183	C₂×C₁₆	—	2	1	no	—
184	C₂×C₁₆	—	2	1	no	—
185	C₂×C₁₆	—	2	1	no	—
186	C₂×C₁₆	D₂₆₈	1	2	yes	—
187	C₂×C₁₆	D₂₆₈	1	2	yes	—
188	#107	—	134	1	no	—
189	#107	—	134	1	no	—
190	#107	—	134	1	no	—
191	#107	—	134	1	no	—
192	#107	—	134	1	no	—
193	#107	—	134	1	no	—
194	#106	—	268	2	no	—
195	#102	—	268	2	no	—
196	C₄×C₈	—	2	1	no	—
197	C₄×C₈	—	2	1	no	—
198	#95	—	268	2	no	—
199	C₂⁴	—	268	1	no	—
200	C₂×D₄	—	268	1	no	—
201	C₂×D₄	—	268	1	no	—
202	C₂×D₄	—	268	1	no	—
203	C₂×D₄	—	268	1	no	—
204	C₂×D₄	—	268	1	no	—
205	C₂×D₄	—	134	2	no	—
206	C₂²×C₄	—	268	1	no	—
207	C₂²×C₄	—	268	1	no	—
208	C₂²×C₄	—	268	1	no	—
209	C₂²×C₄	D₅₃₆	1	2	yes	—
210	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
211	D₈	—	268	1	no	—
212	D₈	—	268	1	no	—
213	D₈	—	134	2	no	—
214	D₈	—	134	2	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	D₅₃₆	1	2	yes	—
218	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
219	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
220	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
221	C₂×C₈	#11761627	1	1	yes	characteristic
222	C₂×C₈	#11761627	1	1	yes	characteristic
223	C₂×C₈	#11761627	1	1	yes	characteristic
224	C₄⋊C₄	—	268	1	no	—
225	C₄⋊C₄	—	268	1	no	—
226	C₄⋊C₄	—	268	1	no	—
227	C₄⋊C₄	—	268	1	no	—
228	C₄⋊C₄	—	268	1	no	—
229	C₄⋊C₄	—	268	1	no	—
230	C₂²⋊C₄	—	536	4	no	—
231	C₄²	—	2	1	no	—
232	C₄²	—	2	1	no	—
233	C₁₆	—	2	2	no	—
234	C₁₆	—	2	2	no	—
235	C₂³	—	268	2	no	—
236	C₂³	—	536	1	no	—
237	C₂³	—	536	2	no	—
238	C₂³	—	536	2	no	—
239	C₂³	—	536	1	no	—
240	C₂³	C₂×D₅₃₆	1	1	yes	characteristic
241	D₄	—	268	2	no	—
242	D₄	—	268	2	no	—
243	D₄	—	536	1	no	—
244	D₄	—	536	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	536	1	no	—
249	C₂×C₄	—	536	1	no	—
250	C₂×C₄	—	536	1	no	—
251	C₂×C₄	—	536	2	no	—
252	C₂×C₄	—	536	2	no	—
253	C₂×C₄	—	536	2	no	—
254	C₂×C₄	#14524023	1	1	yes	characteristic
255	C₂×C₄	#14524027	1	1	yes	characteristic
256	C₂×C₄	#14524040	1	1	yes	characteristic
257	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
258	C₂×C₄	#14524076	1	1	yes	characteristic
259	C₂×C₄	#14524076	1	1	yes	characteristic
260	C₈	—	2	2	no	—
261	C₈	#14524023	1	1	yes	characteristic
262	C₈	#14524023	1	1	yes	characteristic
263	C₈	#14524027	1	1	yes	characteristic
264	C₈	#14524027	1	1	yes	characteristic
265	C₂²	—	536	4	no	—
266	C₂²	—	536	2	no	—
267	C₂²	—	1072	2	no	—
268	C₂²	—	1072	1	no	—
269	C₂²	—	1072	1	no	—
270	C₂²	#16863677	1	1	yes	characteristic
271	C₂²	#16863710	1	1	yes	characteristic
272	C₂²	#16863773	1	1	yes	characteristic
273	C₂²	#16863828	1	2	yes	—
274	C₂²	#16863834	1	2	yes	—
275	C₄	—	4	2	no	—
276	C₄	—	1072	1	no	—
277	C₄	—	1072	1	no	—
278	C₄	—	1072	1	no	—
279	C₄	#16863333	1	1	yes	characteristic
280	C₄	#16863677	1	1	yes	characteristic
281	C₄	#16863710	1	1	yes	characteristic
282	C₄	#16863790	1	1	yes	characteristic
283	C₂	—	1072	2	no	—
284	C₂	#22561736	1	1	yes	characteristic
285	C₂	#22562367	1	2	yes	—
286	C₂	#22562401	1	2	yes	—
287	C₂	#22562470	1	1	yes	characteristic
288	C₂	#22562475	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₃₆
commutator, cosocle	C₂²×C₈×C₆₇	C₂³
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1374	C₂
socle	C₂³×C₆₇	C₂×D₈
prime radical	C₂²×D₅₃₆	C₂²
hypercenter	#1374	D₆₇
hypocenter	C₆₇	#13528
LCS	G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
1. C₂×#16863224	C₂	semidirect
2. C₂×#16863222	C₂	semidirect
6. #22560373	C₂	semidirect
7. C₆₇×#1374	C₂	semidirect
151. #1374	D₆₇	semidirect
152. C₆₇	#13528	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863224	2
C₂×#16863222	2
C₂×#16863198	2
#22561075	2
#22560373	2
C₆₇×#1374	2
#13528	67