Subgroups — C₂×#22562438 #31011121

Order 17152.

622 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863776	C₂	1	1	yes	maximal, characteristic
2	#22562438	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16863199	C₂	1	4	yes	maximal
4	C₂×C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂×#500×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₆₈	—	2	1	no	—
7	C₂²×Dic₂₆₈	C₂²	1	2	yes	—
8	C₂×#14524043	—	2	1	no	—
9	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
10	#16863776	—	2	2	no	—
11	#16863776	C₂²	1	4	yes	—
12	#16863199	C₂²	1	16	yes	—
13	C₄×C₆₇⋊C₁₆	C₂²	1	4	yes	—
14	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
15	#500×C₆₇	—	2	2	no	—
16	#500×C₆₇	C₂²	1	4	yes	—
17	C₂×#107×C₆₇	—	2	1	no	—
18	C₂×#107×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×Dic₁₃₄	—	2	2	no	—
21	C₂×Dic₂₆₈	—	2	2	no	—
22	C₂×Dic₂₆₈	—	2	4	no	—
23	C₂×Dic₂₆₈	—	2	8	no	—
24	C₂×Dic₂₆₈	—	4	4	no	—
25	C₂×Dic₂₆₈	C₂³	1	4	yes	—
26	#14524043	—	2	4	no	—
27	#14524043	—	4	2	no	—
28	C₂×C₄⋊Dic₆₇	—	4	1	no	—
29	C₂×C₄⋊Dic₆₇	—	4	1	no	—
30	C₂×C₆₇⋊C₁₆	—	2	4	no	—
31	C₂×C₆₇⋊C₁₆	C₂³	1	4	yes	—
32	C₂²×Q₈×C₆₇	—	4	1	no	—
33	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
35	C₄⋊Q₈×C₆₇	—	2	4	no	—
36	C₄⋊Q₈×C₆₇	—	4	2	no	—
37	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
39	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
40	#107×C₆₇	—	2	2	no	—
41	#107×C₆₇	—	2	4	no	—
42	#107×C₆₇	C₂³	1	4	yes	—
43	C₄×C₈×C₆₇	D₄	1	2	yes	—
44	C₄×C₈×C₆₇	C₂³	1	2	yes	—
45	C₂×Dic₁₃₄	—	2	4	no	—
46	C₂×Dic₁₃₄	—	4	4	no	—
47	C₂×Dic₁₃₄	—	4	4	no	—
48	C₂×Dic₁₃₄	—	4	2	no	—
49	C₂²×Dic₆₇	—	4	2	no	—
50	Dic₂₆₈	—	2	8	no	—
51	Dic₂₆₈	—	4	4	no	—
52	Dic₂₆₈	—	4	8	no	—
53	C₄⋊Dic₆₇	—	4	4	no	—
54	C₄⋊Dic₆₇	—	4	4	no	—
55	C₆₇⋊C₁₆	—	2	4	no	—
56	C₂×Q₈×C₆₇	—	4	4	no	—
57	C₂×Q₈×C₆₇	—	8	4	no	—
58	C₂²×C₄×C₆₇	—	4	2	no	—
59	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₆₇	SD₁₆	1	4	yes	—
63	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
65	C₂×C₈×C₆₇	#41	1	4	yes	—
66	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
67	C₄⋊C₄×C₆₇	—	2	4	no	—
68	C₄⋊C₄×C₆₇	—	4	2	no	—
69	C₄⋊C₄×C₆₇	—	4	4	no	—
70	C₄²×C₆₇	D₈	1	2	yes	—
71	C₄²×C₆₇	C₂×D₄	1	2	yes	—
72	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
73	Q₈×C₆₇	—	8	8	no	—
74	C₂×C₄×C₆₇	—	4	4	no	—
75	C₂×C₄×C₆₇	—	8	4	no	—
76	C₂×C₄×C₆₇	SD₃₂	1	4	yes	—
77	C₂×C₄×C₆₇	Q₃₂	1	4	yes	—
78	C₂×C₄×C₆₇	#124	1	2	yes	—
79	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
81	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
82	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
83	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
84	C₂×Dic₆₇	—	4	4	no	—
85	C₂×Dic₆₇	—	8	4	no	—
86	Dic₁₃₄	—	4	4	no	—
87	Dic₁₃₄	—	4	4	no	—
88	Dic₁₃₄	—	8	4	no	—
89	C₈×C₆₇	#124	1	2	yes	—
90	C₈×C₆₇	C₂×SD₁₆	1	4	yes	—
91	C₈×C₆₇	C₂×#41	1	2	yes	—
92	C₂²×C₆₇	#486	1	2	yes	—
93	C₂²×C₆₇	C₂×SD₃₂	1	2	yes	—
94	C₂²×C₆₇	C₂×Q₃₂	1	2	yes	—
95	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
96	C₄×C₆₇	—	8	4	no	—
97	C₄×C₆₇	#486	1	2	yes	—
98	C₄×C₆₇	C₂×SD₃₂	1	4	yes	—
99	C₄×C₆₇	C₂×Q₃₂	1	4	yes	—
100	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
101	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
102	Dic₆₇	—	8	4	no	—
103	C₂×#2241	—	67	1	no	maximal, Sylow
104	C₂×C₆₇	#2241	1	4	yes	complemented
105	C₂×C₆₇	C₂×#486	1	1	yes	characteristic
106	C₂×C₆₇	C₂²×SD₃₂	1	1	yes	characteristic
107	C₂×C₆₇	C₂²×Q₃₂	1	1	yes	characteristic
108	C₂×#500	D₆₇	1	1	yes	characteristic
109	C₂×#494	—	67	1	no	—
110	#2241	—	67	8	no	—
111	C₂×#358	—	67	4	no	—
112	C₂×C₄×C₁₆	—	67	1	no	—
113	C₆₇	C₂×#2241	1	1	yes	characteristic, complemented, Sylow
114	C₂²×Q₁₆	—	67	2	no	—
115	C₂²×Q₁₆	—	134	1	no	—
116	C₂×C₄⋊Q₈	—	2	1	no	—
117	C₂×C₄⋊Q₈	—	134	1	no	—
118	C₂²×C₁₆	—	67	2	no	—
119	#500	—	2	2	no	—
120	#500	C₂×D₆₇	1	4	yes	—
121	#494	—	67	4	no	—
122	#494	—	134	2	no	—
123	C₂×#107	—	2	1	no	—
124	C₂×#107	C₂×D₆₇	1	2	yes	—
125	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
126	#358	—	67	16	no	—
127	C₄×C₁₆	—	67	4	no	—
128	C₂²×Q₈	—	4	1	no	—
129	C₂²×Q₈	—	134	2	no	—
130	C₂×Q₁₆	—	268	4	no	—
131	C₂×Q₁₆	—	67	4	no	—
132	C₂×Q₁₆	—	134	2	no	—
133	C₂×Q₁₆	—	134	4	no	—
134	C₂×Q₁₆	—	134	8	no	—
135	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
136	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
137	C₄⋊Q₈	—	2	4	no	—
138	C₄⋊Q₈	—	4	2	no	—
139	C₄⋊Q₈	—	268	2	no	—
140	C₄⋊Q₈	—	134	4	no	—
141	C₂×C₄⋊C₄	—	2	2	no	—
142	C₂×C₄⋊C₄	—	4	1	no	—
143	C₂×C₄⋊C₄	—	268	1	no	—
144	C₂×C₄⋊C₄	—	268	1	no	—
145	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
146	C₂×C₁₆	—	67	4	no	—
147	C₂×C₁₆	—	134	4	no	—
148	#107	—	2	4	no	—
149	#107	—	2	2	no	—
150	#107	C₂²×D₆₇	1	4	yes	—
151	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
152	C₄×C₈	C₂²×D₆₇	1	2	yes	—
153	C₂×Q₈	—	4	4	no	—
154	C₂×Q₈	—	8	4	no	—
155	C₂×Q₈	—	268	2	no	—
156	C₂×Q₈	—	268	4	no	—
157	C₂×Q₈	—	268	4	no	—
158	C₂×Q₈	—	134	4	no	—
159	C₂²×C₄	—	4	2	no	—
160	C₂²×C₄	—	268	2	no	—
161	C₂²×C₄	#11761613	1	1	yes	characteristic
162	C₂²×C₄	#11761613	1	1	yes	characteristic
163	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
164	Q₁₆	—	268	4	no	—
165	Q₁₆	—	268	8	no	—
166	Q₁₆	—	134	8	no	—
167	C₂×C₈	#11761615	1	4	yes	—
168	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
169	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
170	C₂×C₈	#11761629	1	4	yes	—
171	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄⋊C₄	—	4	4	no	—
175	C₄⋊C₄	—	268	4	no	—
176	C₄⋊C₄	—	268	4	no	—
177	C₄²	#11761613	1	2	yes	—
178	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
179	C₁₆	—	134	4	no	—
180	C₂³	C₂×#11761613	1	1	yes	characteristic
181	Q₈	—	8	8	no	—
182	Q₈	—	268	4	no	—
183	Q₈	—	268	4	no	—
184	Q₈	—	536	4	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	—	8	4	no	—
187	C₂×C₄	—	268	4	no	—
188	C₂×C₄	—	536	4	no	—
189	C₂×C₄	#14523987	1	4	yes	—
190	C₂×C₄	#14523989	1	4	yes	—
191	C₂×C₄	#14524032	1	2	yes	—
192	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
193	C₂×C₄	C₂×#11761613	1	2	yes	—
194	C₂×C₄	C₂×#11761613	1	2	yes	—
195	C₂×C₄	C₂×#11761615	1	2	yes	—
196	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
197	C₈	#14524032	1	2	yes	—
198	C₈	C₂×#11761615	1	4	yes	—
199	C₈	C₂×#11761629	1	2	yes	—
200	C₂²	#16863744	1	2	yes	—
201	C₂²	C₂×#14523987	1	2	yes	—
202	C₂²	C₂×#14523989	1	2	yes	—
203	C₂²	C₂²×#11761613	1	1	yes	characteristic
204	C₄	—	8	4	no	—
205	C₄	—	536	4	no	—
206	C₄	#16863744	1	2	yes	—
207	C₄	C₂×#14523987	1	4	yes	—
208	C₄	C₂×#14523989	1	4	yes	—
209	C₄	C₂×#14524032	1	1	yes	characteristic
210	C₄	C₂²×#11761615	1	1	yes	characteristic
211	C₂	#22562438	1	4	yes	direct product factor
212	C₂	C₂×#16863744	1	1	yes	characteristic
213	C₂	C₂²×#14523987	1	1	yes	characteristic
214	C₂	C₂²×#14523989	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11761613
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#500×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×D₈
hypercenter	C₂×#500	D₆₇
hypocenter	C₆₇	C₂×#2241
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
2. #22562438	C₂	direct (N × H)
104. C₂×C₆₇	#2241	semidirect
113. C₆₇	C₂×#2241	semidirect
211. C₂	#22562438	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×#16863776	2
#22562438	2
C₂×#16863199	2
C₂×C₄×C₆₇⋊C₁₆	2
C₂×#500×C₆₇	2
C₂×#2241	67

Subgroups — C2×#22562438 #31011121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562438 #31011121