Subgroups — #31011340

Order 17152.

474 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570748	C₂	1	2	yes	maximal
2	#22562495	C₂	1	4	yes	maximal, complemented
3	#22562488	C₂	1	2	yes	maximal, complemented
4	#22562484	C₂	1	2	yes	maximal, complemented
5	#22561997	C₂	1	2	yes	maximal
6	#22561993	C₂	1	1	yes	maximal, characteristic
7	C₂×#16863222	C₂	1	1	yes	maximal, characteristic
8	C₆₇×#2108	C₂	1	1	yes	maximal, characteristic
9	#16864024	—	2	2	no	—
10	#16863805	—	2	4	no	—
11	#16863801	C₂²	1	4	yes	—
12	#16863798	C₂²	1	4	yes	—
13	#16863438	—	2	2	no	—
14	#16863434	C₂²	1	2	yes	—
15	C₂×#14523977	C₂²	1	2	yes	—
16	#16863224	C₂²	1	4	yes	—
17	#16863224	C₂²	1	2	yes	—
18	#16863222	C₂²	1	1	yes	characteristic
19	#16863222	C₂²	1	2	yes	—
20	#16863222	C₂²	1	4	yes	—
21	#16863222	C₂²	1	1	yes	characteristic
22	#16863222	C₂²	1	2	yes	—
23	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
25	#405×C₆₇	C₂²	1	1	yes	characteristic
26	#346×C₆₇	C₂²	1	2	yes	—
27	C₄×C₁₆×C₆₇	C₂²	1	2	yes	—
28	#14524043	—	2	4	no	—
29	#14524042	—	4	2	no	—
30	#14524035	—	2	4	no	—
31	#14524034	—	4	2	no	—
32	#14524020	—	4	4	no	—
33	#14524002	—	2	2	no	—
34	C₂×C₄⋊Dic₆₇	—	2	2	no	—
35	#14523977	—	2	4	no	—
36	#14523977	—	2	2	no	—
37	#14523977	C₂³	1	2	yes	—
38	#14523977	C₂³	1	4	yes	—
39	#14523977	C₂³	1	2	yes	—
40	#14523975	—	2	4	no	—
41	#14523975	—	2	2	no	—
42	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
44	#117×C₆₇	D₄	1	1	yes	characteristic
45	M₃₂×C₆₇	Q₈	1	4	yes	complemented
46	C₂×C₁₆×C₆₇	Q₈	1	4	yes	complemented
47	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
48	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
49	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
50	#97×C₆₇	D₄	1	2	yes	—
51	C₄×C₈×C₆₇	C₂³	1	2	yes	—
52	C₂×Dic₁₃₄	—	4	4	no	—
53	C₂²×Dic₆₇	—	4	2	no	—
54	C₄⋊Dic₆₇	—	2	2	no	—
55	C₄⋊Dic₆₇	—	2	4	no	—
56	C₄⋊Dic₆₇	—	2	2	no	—
57	C₄⋊Dic₆₇	—	4	2	no	—
58	C₄⋊Dic₆₇	—	4	4	no	—
59	Dic₆₇⋊C₄	—	4	4	no	—
60	Dic₆₇⋊C₄	—	4	4	no	—
61	C₂²⋊Dic₆₇	—	4	2	no	—
62	C₄×Dic₆₇	—	4	2	no	—
63	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
64	M₁₆×C₆₇	Q₁₆	1	4	yes	—
65	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
68	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
69	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
70	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
71	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
72	C₂²⋊C₄×C₆₇	D₈	1	2	yes	—
73	C₄²×C₆₇	C₂×D₄	1	2	yes	—
74	C₁₆×C₆₇	C₂×Q₈	1	2	yes	—
75	C₁₆×C₆₇	C₂×Q₈	1	2	yes	—
76	C₁₆×C₆₇	C₂×Q₈	1	4	yes	—
77	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
79	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
80	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
81	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
83	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
84	C₂×Dic₆₇	—	4	2	no	—
85	C₂×Dic₆₇	—	4	2	no	—
86	C₂×Dic₆₇	—	4	4	no	—
87	C₂×Dic₆₇	—	16	2	no	—
88	Dic₁₃₄	—	16	4	no	—
89	C₈×C₆₇	C₄⋊Q₈	1	2	yes	—
90	C₈×C₆₇	C₂×Q₁₆	1	4	yes	—
91	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
92	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
93	C₂²×C₆₇	—	2	1	no	—
94	C₂²×C₆₇	—	2	1	no	—
95	C₂²×C₆₇	#500	1	2	yes	—
96	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
97	C₄×C₆₇	—	2	2	no	—
98	C₄×C₆₇	—	2	2	no	—
99	C₄×C₆₇	#500	1	2	yes	—
100	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
101	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
102	Dic₆₇	—	8	2	no	—
103	Dic₆₇	—	8	4	no	—
104	Dic₆₇	—	8	2	no	—
105	#33729	—	67	1	no	maximal, Sylow
106	C₂×C₆₇	—	2	2	no	—
107	C₂×C₆₇	C₂×#500	1	1	yes	characteristic
108	C₂×C₆₇	#3415	1	1	yes	characteristic
109	C₂×C₆₇	#3417	1	1	yes	characteristic
110	#3164	—	67	2	no	—
111	#2255	—	67	4	no	—
112	#2253	—	67	2	no	—
113	#2252	—	67	2	no	—
114	#2159	—	67	2	no	—
115	#2158	—	67	1	no	—
116	C₂×#366	—	67	1	no	—
117	#2108	D₆₇	1	1	yes	characteristic
118	C₆₇	#33729	1	1	yes	characteristic, complemented, Sylow
119	#544	—	134	2	no	—
120	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
121	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
122	#501	—	134	4	no	—
123	#500	—	67	4	no	—
124	#499	—	67	4	no	—
125	#428	—	134	2	no	—
126	#427	—	67	2	no	—
127	C₂×#107	—	67	2	no	—
128	#405	C₂×D₆₇	1	1	yes	characteristic
129	#367	—	67	2	no	—
130	#367	—	67	4	no	—
131	#366	—	67	2	no	—
132	#366	—	67	2	no	—
133	#366	—	67	1	no	—
134	#366	—	67	1	no	—
135	#366	—	67	4	no	—
136	#346	C₂×D₆₇	1	2	yes	—
137	C₄×C₁₆	C₂×D₆₇	1	2	yes	—
138	C₂×M₁₆	D₂₆₈	1	1	yes	characteristic
139	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
140	C₄⋊Q₈	—	268	2	no	—
141	C₄⋊Q₈	—	134	4	no	—
142	#125	—	268	2	no	—
143	#125	—	134	4	no	—
144	C₂²⋊Q₈	—	268	4	no	—
145	#117	—	134	2	no	—
146	#117	D₂₆₈	1	1	yes	characteristic
147	C₂×C₄⋊C₄	—	134	2	no	—
148	M₃₂	Dic₁₃₄	1	4	yes	complemented
149	C₂×C₁₆	Dic₁₃₄	1	4	yes	complemented
150	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
151	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
152	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
153	#107	—	67	4	no	—
154	#107	—	67	2	no	—
155	#107	—	67	2	no	—
156	#107	—	134	2	no	—
157	#107	—	134	4	no	—
158	#106	—	134	2	no	—
159	#106	—	134	4	no	—
160	#97	D₂₆₈	1	2	yes	—
161	C₄×C₈	C₂²×D₆₇	1	2	yes	—
162	C₂×Q₈	—	268	4	no	—
163	C₂²×C₄	—	268	2	no	—
164	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
165	M₁₆	Dic₂₆₈	1	4	yes	—
166	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
167	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
168	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
169	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
170	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
171	C₄⋊C₄	—	268	2	no	—
172	C₄⋊C₄	—	268	4	no	—
173	C₄⋊C₄	—	268	4	no	—
174	C₄⋊C₄	—	268	4	no	—
175	C₄⋊C₄	—	134	2	no	—
176	C₄⋊C₄	—	134	4	no	—
177	C₄⋊C₄	—	134	2	no	—
178	C₄⋊C₄	D₅₃₆	1	1	yes	characteristic
179	C₄⋊C₄	D₅₃₆	1	1	yes	characteristic
180	C₂²⋊C₄	—	268	2	no	—
181	C₂²⋊C₄	D₅₃₆	1	2	yes	—
182	C₄²	—	268	2	no	—
183	C₄²	C₂×D₂₆₈	1	2	yes	—
184	C₁₆	C₂×Dic₁₃₄	1	2	yes	—
185	C₁₆	C₂×Dic₁₃₄	1	2	yes	—
186	C₁₆	C₂×Dic₁₃₄	1	4	yes	—
187	C₂³	C₂×D₅₃₆	1	1	yes	characteristic
188	Q₈	—	1072	4	no	—
189	C₂×C₄	—	268	2	no	—
190	C₂×C₄	—	268	2	no	—
191	C₂×C₄	—	268	4	no	—
192	C₂×C₄	—	1072	2	no	—
193	C₂×C₄	#14524043	1	2	yes	—
194	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
195	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
196	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
197	C₂×C₄	C₂×Dic₂₆₈	1	2	yes	—
198	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
199	C₈	#14524043	1	2	yes	—
200	C₈	C₂×Dic₂₆₈	1	4	yes	—
201	C₈	C₂²×Dic₁₃₄	1	1	yes	characteristic
202	C₈	C₂²×Dic₁₃₄	1	1	yes	characteristic
203	C₂²	—	2	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	#16863801	1	2	yes	—
206	C₂²	C₂²×D₅₃₆	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	536	4	no	—
210	C₄	—	536	2	no	—
211	C₄	—	536	2	no	—
212	C₄	#16863801	1	2	yes	—
213	C₄	C₂×#14524043	1	1	yes	characteristic
214	C₄	C₂²×Dic₂₆₈	1	1	yes	characteristic
215	C₂	—	2	2	no	—
216	C₂	C₂×#16863801	1	1	yes	characteristic
217	C₂	#22573350	1	1	yes	characteristic
218	C₂	#22573362	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₅₃₆
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#2108	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂³×C₆₇	C₂×D₈
hypercenter	#2108	D₆₇
hypocenter	C₆₇	#33729
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
2. #22562495	C₂	semidirect
3. #22562488	C₂	semidirect
4. #22562484	C₂	semidirect
45. M₃₂×C₆₇	Q₈	semidirect
46. C₂×C₁₆×C₆₇	Q₈	semidirect
118. C₆₇	#33729	semidirect
148. M₃₂	Dic₁₃₄	semidirect
149. C₂×C₁₆	Dic₁₃₄	semidirect

Maximal subgroups

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

subgroup	index
#22570748	2
#22562495	2
#22562488	2
#22562484	2
#22561997	2
#22561993	2
C₂×#16863222	2
C₆₇×#2108	2
#33729	67