Subgroups — #31011348

Order 17152.

474 conjugacy classes of subgroups.

The subgroup lattice has 210 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	#22562491	C₂	1	2	yes	maximal, complemented
4	#22562488	C₂	1	2	yes	maximal, complemented
5	#22561997	C₂	1	2	yes	maximal
6	#22561993	C₂	1	1	yes	maximal, characteristic
7	C₂×#16863224	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	1	yes	characteristic
17	#16863224	C₂²	1	2	yes	—
18	#16863224	C₂²	1	1	yes	characteristic
19	#16863224	C₂²	1	4	yes	—
20	#16863224	C₂²	1	2	yes	—
21	#16863222	C₂²	1	4	yes	—
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	2	yes	—
71	C₂²⋊C₄×C₆₇	D₈	1	2	yes	—
72	C₄²×C₆₇	C₂×D₄	1	2	yes	—
73	C₁₆×C₆₇	C₂×Q₈	1	2	yes	—
74	C₁₆×C₆₇	C₂×Q₈	1	2	yes	—
75	C₁₆×C₆₇	C₂×Q₈	1	4	yes	—
76	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
77	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
78	C₂×C₄×C₆₇	C₂×D₈	1	4	yes	—
79	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
81	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
82	C₂×Dic₆₇	—	4	2	no	—
83	C₂×Dic₆₇	—	4	2	no	—
84	C₂×Dic₆₇	—	4	4	no	—
85	C₂×Dic₆₇	—	16	2	no	—
86	Dic₁₃₄	—	16	4	no	—
87	C₈×C₆₇	C₄⋊Q₈	1	2	yes	—
88	C₈×C₆₇	C₂×Q₁₆	1	4	yes	—
89	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
90	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
91	C₂²×C₆₇	—	2	2	no	—
92	C₂²×C₆₇	#500	1	2	yes	—
93	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
94	C₄×C₆₇	—	2	4	no	—
95	C₄×C₆₇	#500	1	2	yes	—
96	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
97	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
98	Dic₆₇	—	8	2	no	—
99	Dic₆₇	—	8	4	no	—
100	Dic₆₇	—	8	2	no	—
101	#33730	—	67	1	no	maximal, Sylow
102	C₂×C₆₇	—	2	2	no	—
103	C₂×C₆₇	C₂×#500	1	1	yes	characteristic
104	C₂×C₆₇	#3416	1	2	yes	—
105	#3164	—	67	2	no	—
106	#2255	—	67	4	no	—
107	#2254	—	67	2	no	—
108	#2253	—	67	2	no	—
109	#2159	—	67	2	no	—
110	#2158	—	67	1	no	—
111	C₂×#367	—	67	1	no	—
112	#2108	D₆₇	1	1	yes	characteristic
113	C₆₇	#33730	1	1	yes	characteristic, complemented, Sylow
114	#544	—	134	2	no	—
115	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
116	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
117	#501	—	134	4	no	—
118	#500	—	67	4	no	—
119	#499	—	67	4	no	—
120	#428	—	134	2	no	—
121	#427	—	67	2	no	—
122	C₂×#107	—	67	2	no	—
123	#405	C₂×D₆₇	1	1	yes	characteristic
124	#367	—	67	2	no	—
125	#367	—	67	2	no	—
126	#367	—	67	1	no	—
127	#367	—	67	1	no	—
128	#367	—	67	4	no	—
129	#366	—	67	2	no	—
130	#366	—	67	4	no	—
131	#346	C₂×D₆₇	1	2	yes	—
132	C₄×C₁₆	C₂×D₆₇	1	2	yes	—
133	C₂×M₁₆	D₂₆₈	1	1	yes	characteristic
134	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
135	C₄⋊Q₈	—	268	2	no	—
136	C₄⋊Q₈	—	134	4	no	—
137	#125	—	268	2	no	—
138	#125	—	134	4	no	—
139	C₂²⋊Q₈	—	268	4	no	—
140	#117	—	134	2	no	—
141	#117	D₂₆₈	1	1	yes	characteristic
142	C₂×C₄⋊C₄	—	134	2	no	—
143	M₃₂	Dic₁₃₄	1	4	yes	complemented
144	C₂×C₁₆	Dic₁₃₄	1	4	yes	complemented
145	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
146	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
147	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
148	#107	—	67	2	no	—
149	#107	—	67	4	no	—
150	#107	—	67	2	no	—
151	#107	—	134	2	no	—
152	#107	—	134	4	no	—
153	#106	—	134	2	no	—
154	#106	—	134	4	no	—
155	#97	D₂₆₈	1	2	yes	—
156	C₄×C₈	C₂²×D₆₇	1	2	yes	—
157	C₂×Q₈	—	268	4	no	—
158	C₂²×C₄	—	268	2	no	—
159	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
160	M₁₆	Dic₂₆₈	1	4	yes	—
161	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
162	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
163	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
164	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
165	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
166	C₄⋊C₄	—	268	2	no	—
167	C₄⋊C₄	—	268	4	no	—
168	C₄⋊C₄	—	268	4	no	—
169	C₄⋊C₄	—	268	4	no	—
170	C₄⋊C₄	—	134	2	no	—
171	C₄⋊C₄	—	134	4	no	—
172	C₄⋊C₄	—	134	2	no	—
173	C₄⋊C₄	D₅₃₆	1	2	yes	—
174	C₂²⋊C₄	—	268	2	no	—
175	C₂²⋊C₄	D₅₃₆	1	2	yes	—
176	C₄²	—	268	2	no	—
177	C₄²	C₂×D₂₆₈	1	2	yes	—
178	C₁₆	C₂×Dic₁₃₄	1	2	yes	—
179	C₁₆	C₂×Dic₁₃₄	1	2	yes	—
180	C₁₆	C₂×Dic₁₃₄	1	4	yes	—
181	C₂³	C₂×D₅₃₆	1	1	yes	characteristic
182	Q₈	—	1072	4	no	—
183	C₂×C₄	—	268	2	no	—
184	C₂×C₄	—	268	2	no	—
185	C₂×C₄	—	268	4	no	—
186	C₂×C₄	—	1072	2	no	—
187	C₂×C₄	#14524043	1	2	yes	—
188	C₂×C₄	C₂×D₅₃₆	1	4	yes	—
189	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
190	C₂×C₄	C₂×Dic₂₆₈	1	2	yes	—
191	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
192	C₈	#14524043	1	2	yes	—
193	C₈	C₂×Dic₂₆₈	1	4	yes	—
194	C₈	C₂²×Dic₁₃₄	1	1	yes	characteristic
195	C₈	C₂²×Dic₁₃₄	1	1	yes	characteristic
196	C₂²	—	2	2	no	—
197	C₂²	#16863801	1	2	yes	—
198	C₂²	C₂²×D₅₃₆	1	1	yes	characteristic
199	C₄	—	2	4	no	—
200	C₄	—	536	4	no	—
201	C₄	—	536	2	no	—
202	C₄	—	536	2	no	—
203	C₄	#16863801	1	2	yes	—
204	C₄	C₂×#14524043	1	1	yes	characteristic
205	C₄	C₂²×Dic₂₆₈	1	1	yes	characteristic
206	C₂	—	2	2	no	—
207	C₂	C₂×#16863801	1	1	yes	characteristic
208	C₂	#22573357	1	2	yes	—

	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	#33730
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. #22562491	C₂	semidirect
4. #22562488	C₂	semidirect
45. M₃₂×C₆₇	Q₈	semidirect
46. C₂×C₁₆×C₆₇	Q₈	semidirect
113. C₆₇	#33730	semidirect
143. M₃₂	Dic₁₃₄	semidirect
144. 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
#22562491	2
#22562488	2
#22561997	2
#22561993	2
C₂×#16863224	2
C₆₇×#2108	2
#33730	67