Subgroups — C₂×#22562485 #31011310

Order 17152.

494 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863798	C₂	1	1	yes	maximal, characteristic
2	#22562485	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16863223	C₂	1	2	yes	maximal
4	C₂×#16863221	C₂	1	2	yes	maximal
5	C₂×C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂×#499×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#14524035	—	2	1	no	—
8	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
10	#16863798	—	2	2	no	—
11	#16863798	C₂²	1	4	yes	—
12	C₂×#14523977	C₂²	1	2	yes	—
13	C₂×#14523975	—	2	1	no	—
14	#16863223	C₂²	1	8	yes	—
15	#16863221	C₂²	1	8	yes	—
16	C₄×C₆₇⋊C₁₆	C₂²	1	4	yes	—
17	C₂×#125×C₆₇	—	2	1	no	—
18	#499×C₆₇	—	2	2	no	—
19	#499×C₆₇	C₂²	1	4	yes	—
20	C₂×#107×C₆₇	C₂²	1	2	yes	—
21	C₂×#106×C₆₇	—	2	1	no	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	#14524035	—	2	4	no	—
24	#14524035	—	4	2	no	—
25	C₂×C₄⋊Dic₆₇	—	2	2	no	—
26	C₂×Dic₆₇⋊C₄	—	4	2	no	—
27	C₂×C₆₇⋊C₁₆	Q₈	1	4	yes	—
28	C₂×C₆₇⋊C₁₆	Q₈	1	4	yes	—
29	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
30	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
31	#14523977	—	2	2	no	—
32	#14523977	C₂³	1	4	yes	—
33	#14523975	—	2	4	no	—
34	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
36	#125×C₆₇	—	2	4	no	—
37	#125×C₆₇	—	4	2	no	—
38	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
40	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
41	#107×C₆₇	—	2	2	no	—
42	#107×C₆₇	C₂³	1	4	yes	—
43	#106×C₆₇	—	2	4	no	—
44	C₄×C₈×C₆₇	D₄	1	2	yes	—
45	C₄×C₈×C₆₇	C₂³	1	2	yes	—
46	C₂²×Dic₆₇	—	4	2	no	—
47	C₄⋊Dic₆₇	—	2	4	no	—
48	C₄⋊Dic₆₇	—	4	2	no	—
49	Dic₆₇⋊C₄	—	4	8	no	—
50	C₆₇⋊C₁₆	C₂×Q₈	1	4	yes	—
51	C₆₇⋊C₁₆	C₂×Q₈	1	4	yes	—
52	C₂²×C₄×C₆₇	—	4	2	no	—
53	C₂²×C₄×C₆₇	D₈	1	2	yes	—
54	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₆₇	Q₁₆	1	4	yes	—
56	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
59	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
60	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₄×C₆₇	—	2	4	no	—
62	C₄⋊C₄×C₆₇	—	4	2	no	—
63	C₄⋊C₄×C₆₇	—	4	8	no	—
64	C₄²×C₆₇	D₈	1	2	yes	—
65	C₄²×C₆₇	C₂×D₄	1	2	yes	—
66	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₆₇	—	2	4	no	—
68	C₂×C₄×C₆₇	—	4	4	no	—
69	C₂×C₄×C₆₇	—	8	4	no	—
70	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
71	C₂×C₄×C₆₇	C₂×D₈	1	4	yes	—
72	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
74	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
75	C₂×Dic₆₇	—	4	4	no	—
76	C₂×Dic₆₇	—	8	4	no	—
77	C₈×C₆₇	C₄⋊Q₈	1	2	yes	—
78	C₈×C₆₇	C₂×Q₁₆	1	4	yes	—
79	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
80	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
81	C₂²×C₆₇	#500	1	2	yes	—
82	C₂²×C₆₇	#508	1	4	yes	—
83	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
84	C₄×C₆₇	—	2	4	no	—
85	C₄×C₆₇	—	8	4	no	—
86	C₄×C₆₇	#500	1	2	yes	—
87	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
88	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
89	Dic₆₇	—	8	4	no	—
90	C₂×#2253	—	67	1	no	maximal, Sylow
91	C₂×C₆₇	#2253	1	4	yes	complemented
92	C₂×C₆₇	C₂×#500	1	1	yes	characteristic
93	C₂×C₆₇	C₂×#508	1	2	yes	—
94	C₂×#499	—	67	1	no	—
95	C₂×#499	D₆₇	1	1	yes	characteristic
96	#2253	—	67	8	no	—
97	C₂×#367	—	67	2	no	—
98	C₂×#366	—	67	2	no	—
99	C₂×C₄×C₁₆	—	67	1	no	—
100	C₆₇	C₂×#2253	1	1	yes	characteristic, complemented, Sylow
101	C₂×#125	—	2	1	no	—
102	C₂×#125	—	134	1	no	—
103	C₂²×C₁₆	—	67	1	no	—
104	C₂²×C₁₆	—	67	1	no	—
105	#499	—	2	2	no	—
106	#499	—	67	4	no	—
107	#499	—	134	2	no	—
108	#499	C₂×D₆₇	1	4	yes	—
109	C₂×#107	—	67	2	no	—
110	C₂×#107	C₂×D₆₇	1	2	yes	—
111	C₂×#106	—	2	1	no	—
112	C₂×#106	—	134	1	no	—
113	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
114	#367	—	67	8	no	—
115	#366	—	67	8	no	—
116	C₄×C₁₆	—	67	4	no	—
117	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
118	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
119	#125	—	2	4	no	—
120	#125	—	4	2	no	—
121	#125	—	268	2	no	—
122	#125	—	134	4	no	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄⋊C₄	—	4	2	no	—
125	C₂×C₄⋊C₄	—	268	2	no	—
126	C₂×C₄⋊C₄	—	134	2	no	—
127	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
128	C₂×C₁₆	—	67	4	no	—
129	C₂×C₁₆	—	67	4	no	—
130	C₂×C₁₆	—	67	2	no	—
131	C₂×C₁₆	—	67	2	no	—
132	#107	—	2	2	no	—
133	#107	—	67	4	no	—
134	#107	—	134	2	no	—
135	#107	C₂²×D₆₇	1	4	yes	—
136	#106	—	2	4	no	—
137	#106	—	134	4	no	—
138	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
139	C₄×C₈	C₂²×D₆₇	1	2	yes	—
140	C₂²×C₄	—	4	2	no	—
141	C₂²×C₄	—	268	2	no	—
142	C₂²×C₄	#11761613	1	2	yes	—
143	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
144	C₂×C₈	#11761618	1	4	yes	—
145	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
146	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
147	C₂×C₈	Q₈×D₆₇	1	2	yes	—
148	C₂×C₈	Q₈×D₆₇	1	2	yes	—
149	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
150	C₄⋊C₄	—	2	4	no	—
151	C₄⋊C₄	—	4	2	no	—
152	C₄⋊C₄	—	4	8	no	—
153	C₄⋊C₄	—	268	8	no	—
154	C₄⋊C₄	—	268	2	no	—
155	C₄⋊C₄	—	134	4	no	—
156	C₄²	#11761613	1	2	yes	—
157	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
158	C₁₆	—	67	4	no	—
159	C₁₆	—	67	4	no	—
160	C₂³	C₂×#11761613	1	1	yes	characteristic
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	—	8	4	no	—
164	C₂×C₄	—	268	4	no	—
165	C₂×C₄	—	536	4	no	—
166	C₂×C₄	#14524041	1	2	yes	—
167	C₂×C₄	C₂×#11761613	1	4	yes	—
168	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
169	C₂×C₄	C₂×#11761618	1	2	yes	—
170	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
171	C₈	#14524041	1	2	yes	—
172	C₈	C₂×#11761618	1	4	yes	—
173	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
174	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
175	C₂²	#16863799	1	2	yes	—
176	C₂²	#16863826	1	4	yes	—
177	C₂²	C₂²×#11761613	1	1	yes	characteristic
178	C₄	—	2	4	no	—
179	C₄	—	8	4	no	—
180	C₄	—	536	4	no	—
181	C₄	#16863799	1	2	yes	—
182	C₄	C₂×#14524041	1	1	yes	characteristic
183	C₄	C₂²×#11761618	1	1	yes	characteristic
184	C₂	#22562485	1	4	yes	direct product factor
185	C₂	C₂×#16863799	1	1	yes	characteristic
186	C₂	C₂×#16863826	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11761613
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#499×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×D₈
hypercenter	C₂×#499	D₆₇
hypocenter	C₆₇	C₂×#2253
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
2. #22562485	C₂	direct (N × H)
91. C₂×C₆₇	#2253	semidirect
100. C₆₇	C₂×#2253	semidirect
184. C₂	#22562485	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16863798	2
#22562485	2
C₂×#16863223	2
C₂×#16863221	2
C₂×C₄×C₆₇⋊C₁₆	2
C₂×#499×C₆₇	2
C₂×#2253	67

Subgroups — C2×#22562485 #31011310

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562485 #31011310