Subgroups — C₂×#22562496 #31011329

Order 17152.

518 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	C₂×#16863801	C₂	1	1	yes	maximal, characteristic
2	#22562496	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×#16863172	C₂	1	1	yes	maximal, characteristic
6	C₂×#499×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#14524043	—	2	1	no	—
8	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
9	#16863801	—	2	2	no	—
10	#16863801	C₂²	1	4	yes	—
11	C₂×#14523977	—	2	1	no	—
12	C₂×#14523977	C₂²	1	2	yes	—
13	#16863223	C₂²	1	8	yes	—
14	#16863221	C₂²	1	8	yes	—
15	#16863172	C₂²	1	4	yes	—
16	C₂×#125×C₆₇	—	2	1	no	—
17	#499×C₆₇	—	2	2	no	—
18	#499×C₆₇	C₂²	1	4	yes	—
19	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#106×C₆₇	—	2	1	no	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×Dic₁₃₄	—	4	1	no	—
24	#14524043	—	2	4	no	—
25	#14524043	—	4	2	no	—
26	C₂×C₄⋊Dic₆₇	—	2	2	no	—
27	C₂×C₄⋊Dic₆₇	—	4	1	no	—
28	C₂×C₆₇⋊C₁₆	Q₈	1	8	yes	complemented
29	C₂×C₆₇⋊C₁₆	C₂³	1	4	yes	—
30	#14523977	—	2	2	no	—
31	#14523977	—	2	4	no	—
32	#14523977	C₂³	1	4	yes	—
33	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
35	#125×C₆₇	—	2	4	no	—
36	#125×C₆₇	—	4	2	no	—
37	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
38	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
39	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
40	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
41	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
42	#107×C₆₇	—	2	1	no	—
43	#107×C₆₇	—	2	1	no	—
44	#107×C₆₇	C₂³	1	2	yes	—
45	#107×C₆₇	C₂³	1	2	yes	—
46	#106×C₆₇	—	2	4	no	—
47	C₄×C₈×C₆₇	D₄	1	2	yes	—
48	C₄×C₈×C₆₇	C₂³	1	2	yes	—
49	C₂×Dic₁₃₄	—	4	4	no	—
50	C₂×Dic₁₃₄	—	8	4	no	—
51	C₂²×Dic₆₇	—	4	2	no	—
52	C₄⋊Dic₆₇	—	2	4	no	—
53	C₄⋊Dic₆₇	—	4	2	no	—
54	C₄⋊Dic₆₇	—	4	4	no	—
55	C₆₇⋊C₁₆	C₂×Q₈	1	8	yes	complemented
56	C₂²×C₄×C₆₇	—	4	1	no	—
57	C₂²×C₄×C₆₇	—	4	1	no	—
58	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₆₇	Q₁₆	1	4	yes	—
62	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
64	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
65	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
66	C₄⋊C₄×C₆₇	—	2	2	no	—
67	C₄⋊C₄×C₆₇	—	2	2	no	—
68	C₄⋊C₄×C₆₇	—	4	1	no	—
69	C₄⋊C₄×C₆₇	—	4	1	no	—
70	C₄⋊C₄×C₆₇	—	4	4	no	—
71	C₄⋊C₄×C₆₇	—	4	4	no	—
72	C₄²×C₆₇	D₈	1	2	yes	—
73	C₄²×C₆₇	C₂×D₄	1	2	yes	—
74	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₆₇	—	2	2	no	—
76	C₂×C₄×C₆₇	—	2	2	no	—
77	C₂×C₄×C₆₇	—	4	2	no	—
78	C₂×C₄×C₆₇	—	4	2	no	—
79	C₂×C₄×C₆₇	—	8	2	no	—
80	C₂×C₄×C₆₇	—	8	2	no	—
81	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
82	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
83	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
84	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
86	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
87	C₂×Dic₆₇	—	4	4	no	—
88	C₂×Dic₆₇	—	8	4	no	—
89	Dic₁₃₄	—	8	8	no	—
90	C₈×C₆₇	C₄⋊Q₈	1	2	yes	—
91	C₈×C₆₇	C₂×Q₁₆	1	4	yes	—
92	C₈×C₆₇	C₂²×Q₈	1	2	yes	—
93	C₂²×C₆₇	#500	1	2	yes	—
94	C₂²×C₆₇	#509	1	2	yes	—
95	C₂²×C₆₇	#510	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	—	8	2	no	—
100	C₄×C₆₇	—	8	2	no	—
101	C₄×C₆₇	#500	1	2	yes	—
102	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
103	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
104	Dic₆₇	—	8	4	no	—
105	C₂×#2255	—	67	1	no	maximal, Sylow
106	C₂×C₆₇	#2255	1	4	yes	complemented
107	C₂×C₆₇	C₂×#500	1	1	yes	characteristic
108	C₂×C₆₇	C₂×#509	1	1	yes	characteristic
109	C₂×C₆₇	C₂×#510	1	1	yes	characteristic
110	C₂×#500	—	67	1	no	—
111	C₂×#499	D₆₇	1	1	yes	characteristic
112	#2255	—	67	8	no	—
113	C₂×#367	—	67	2	no	—
114	C₂×#366	—	67	2	no	—
115	C₂×#346	—	67	1	no	—
116	C₆₇	C₂×#2255	1	1	yes	characteristic, complemented, Sylow
117	C₂×C₄⋊Q₈	—	134	1	no	—
118	C₂×#125	—	2	1	no	—
119	C₂²×C₁₆	—	67	2	no	—
120	#500	—	67	4	no	—
121	#500	—	134	2	no	—
122	#499	—	2	2	no	—
123	#499	C₂×D₆₇	1	4	yes	—
124	C₂×#107	—	67	2	no	—
125	C₂×#107	—	134	1	no	—
126	C₂×#107	C₂×D₆₇	1	1	yes	characteristic
127	C₂×#107	C₂×D₆₇	1	1	yes	characteristic
128	C₂×#106	—	2	1	no	—
129	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
130	#367	—	67	8	no	—
131	#366	—	67	8	no	—
132	#346	—	67	4	no	—
133	C₂²×Q₈	—	268	1	no	—
134	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
136	C₄⋊Q₈	—	268	2	no	—
137	C₄⋊Q₈	—	134	4	no	—
138	#125	—	2	4	no	—
139	#125	—	4	2	no	—
140	C₂×C₄⋊C₄	—	2	1	no	—
141	C₂×C₄⋊C₄	—	2	1	no	—
142	C₂×C₄⋊C₄	—	4	1	no	—
143	C₂×C₄⋊C₄	—	4	1	no	—
144	C₂×C₄⋊C₄	—	268	1	no	—
145	C₂×C₄⋊C₄	—	134	2	no	—
146	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
147	C₂×C₁₆	—	67	8	no	—
148	C₂×C₁₆	—	67	4	no	—
149	#107	—	2	1	no	—
150	#107	—	2	1	no	—
151	#107	—	67	4	no	—
152	#107	—	134	2	no	—
153	#107	—	134	4	no	—
154	#107	C₂²×D₆₇	1	2	yes	—
155	#107	C₂²×D₆₇	1	2	yes	—
156	#106	—	2	4	no	—
157	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
158	C₄×C₈	C₂²×D₆₇	1	2	yes	—
159	C₂×Q₈	—	268	4	no	—
160	C₂×Q₈	—	536	4	no	—
161	C₂²×C₄	—	4	1	no	—
162	C₂²×C₄	—	4	1	no	—
163	C₂²×C₄	—	268	2	no	—
164	C₂²×C₄	#11761613	1	1	yes	characteristic
165	C₂²×C₄	#11761613	1	1	yes	characteristic
166	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
167	C₂×C₈	#11761618	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	Q₈×D₆₇	1	4	yes	—
171	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	4	1	no	—
175	C₄⋊C₄	—	4	1	no	—
176	C₄⋊C₄	—	4	4	no	—
177	C₄⋊C₄	—	4	4	no	—
178	C₄⋊C₄	—	268	4	no	—
179	C₄⋊C₄	—	268	2	no	—
180	C₄⋊C₄	—	134	4	no	—
181	C₄²	#11761613	1	2	yes	—
182	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
183	C₁₆	—	67	8	no	—
184	C₂³	C₂×#11761613	1	1	yes	characteristic
185	Q₈	—	536	8	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	8	2	no	—
191	C₂×C₄	—	8	2	no	—
192	C₂×C₄	—	268	4	no	—
193	C₂×C₄	—	536	4	no	—
194	C₂×C₄	#14524041	1	2	yes	—
195	C₂×C₄	C₂×#11761613	1	2	yes	—
196	C₂×C₄	C₂×#11761613	1	2	yes	—
197	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
198	C₂×C₄	C₂×#11761618	1	2	yes	—
199	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
200	C₈	#14524041	1	2	yes	—
201	C₈	C₂×#11761618	1	4	yes	—
202	C₈	C₂×Q₈×D₆₇	1	2	yes	—
203	C₂²	#16863799	1	2	yes	—
204	C₂²	#16863831	1	2	yes	—
205	C₂²	#16863836	1	2	yes	—
206	C₂²	C₂²×#11761613	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	8	2	no	—
210	C₄	—	8	2	no	—
211	C₄	—	536	4	no	—
212	C₄	#16863799	1	2	yes	—
213	C₄	C₂×#14524041	1	1	yes	characteristic
214	C₄	C₂²×#11761618	1	1	yes	characteristic
215	C₂	#22562496	1	4	yes	direct product factor
216	C₂	C₂×#16863799	1	1	yes	characteristic
217	C₂	C₂×#16863831	1	1	yes	characteristic
218	C₂	C₂×#16863836	1	1	yes	characteristic

	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×#2255
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. #22562496	C₂	direct (N × H)
28. C₂×C₆₇⋊C₁₆	Q₈	semidirect
55. C₆₇⋊C₁₆	C₂×Q₈	semidirect
106. C₂×C₆₇	#2255	semidirect
116. C₆₇	C₂×#2255	semidirect
215. C₂	#22562496	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×#16863801	2
#22562496	2
C₂×#16863223	2
C₂×#16863221	2
C₂×#16863172	2
C₂×#499×C₆₇	2
C₂×#2255	67

Subgroups — C2×#22562496 #31011329

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562496 #31011329