Subgroups — C₂×#22569311 #31720795

Order 17152.

1794 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569311	C₂	1	16	yes	maximal, direct product factor
2	C₂×#16863527	C₂	1	4	yes	maximal, complemented
3	C₂×Q₈×C₆₇⋊C₈	C₂	1	4	yes	maximal, complemented
4	C₂×#16863457	C₂	1	4	yes	maximal
5	C₂×#16863449	C₂	1	2	yes	maximal
6	C₂×#533×C₆₇	C₂	1	1	yes	maximal, characteristic
7	#16863527	C₂²	1	32	yes	complemented
8	Q₈×C₆₇⋊C₈	C₂²	1	32	yes	complemented
9	#16863457	C₂²	1	32	yes	—
10	#16863449	C₂²	1	16	yes	—
11	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	8	yes	—
14	C₂×C₂²⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
15	C₂×#14523954	C₂²	1	4	yes	—
16	C₂×C₄×C₆₇⋊C₈	C₂²	1	4	yes	—
17	C₂×C₄×C₆₇⋊C₈	C₂²	1	4	yes	—
18	#533×C₆₇	C₄	1	8	yes	—
19	#533×C₆₇	C₂²	1	8	yes	—
20	C₂×C₄⋊Q₈×C₆₇	C₄	1	2	yes	—
21	C₂×#125×C₆₇	C₄	1	2	yes	—
22	C₂×C₂²⋊Q₈×C₆₇	C₄	1	4	yes	—
23	C₂×C₄×Q₈×C₆₇	C₂²	1	4	yes	—
24	C₂×#117×C₆₇	C₂²	1	2	yes	—
25	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
26	C₂²×C₆₇⋊C₈	C₂³	1	4	yes	—
27	C₂²×C₆₇⋊C₈	C₂³	1	4	yes	—
28	C₄⋊(C₆₇⋊C₈)	C₂³	1	8	yes	—
29	C₄⋊(C₆₇⋊C₈)	C₂³	1	32	yes	—
30	C₄⋊(C₆₇⋊C₈)	C₂³	1	8	yes	—
31	C₂²⋊(C₆₇⋊C₈)	C₂³	1	16	yes	—
32	#14523954	C₂³	1	16	yes	—
33	C₄×C₆₇⋊C₈	C₂³	1	16	yes	—
34	C₄×C₆₇⋊C₈	C₂³	1	16	yes	—
35	C₂²×Q₈×C₆₇	C₂×C₄	1	4	yes	—
36	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
37	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
38	C₄⋊Q₈×C₆₇	C₂×C₄	1	16	yes	—
39	#125×C₆₇	C₂×C₄	1	16	yes	—
40	C₂²⋊Q₈×C₆₇	C₂×C₄	1	32	yes	—
41	C₄×Q₈×C₆₇	C₂×C₄	1	16	yes	—
42	C₄×Q₈×C₆₇	C₂³	1	16	yes	—
43	#117×C₆₇	C₂×C₄	1	8	yes	—
44	#117×C₆₇	C₂³	1	8	yes	—
45	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
46	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	8	yes	—
47	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
48	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
52	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
53	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
54	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
55	C₂×C₆₇⋊C₈	—	2	8	no	—
56	C₂×C₆₇⋊C₈	—	2	8	no	—
57	C₂×C₆₇⋊C₈	C₂⁴	1	8	yes	—
58	C₂×C₆₇⋊C₈	C₂⁴	1	8	yes	—
59	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
60	C₂×Q₈×C₆₇	—	2	16	no	—
61	C₂×Q₈×C₆₇	C₂²×C₄	1	16	yes	—
62	C₂²×C₄×C₆₇	—	2	8	no	—
63	C₂²×C₄×C₆₇	—	2	4	no	—
64	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
68	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
69	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
71	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
72	C₄⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
73	C₄⋊C₄×C₆₇	C₂²×C₄	1	32	yes	—
74	C₄⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
75	C₂²⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
76	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
77	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
78	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
79	C₄²×C₆₇	C₂²×C₄	1	8	yes	—
80	C₄²×C₆₇	C₂⁴	1	4	yes	—
81	C₄²×C₆₇	C₂⁴	1	8	yes	—
82	C₂³×C₆₇	—	2	2	no	—
83	C₂³×C₆₇	—	2	4	no	—
84	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
85	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
86	Q₈×C₆₇	—	2	32	no	—
87	C₂×C₄×C₆₇	—	2	2	no	—
88	C₂×C₄×C₆₇	—	2	4	no	—
89	C₂×C₄×C₆₇	—	2	16	no	—
90	C₂×C₄×C₆₇	—	2	4	no	—
91	C₂×C₄×C₆₇	—	2	16	no	—
92	C₂×C₄×C₆₇	—	2	4	no	—
93	C₂×C₄×C₆₇	#131	1	8	yes	—
94	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
95	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
96	C₂×C₄×C₆₇	C₂³×C₄	1	16	yes	—
97	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
99	C₂×C₄×C₆₇	2₋¹⁺⁴	1	4	yes	—
100	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
101	C₆₇⋊C₈	—	2	8	no	—
102	C₆₇⋊C₈	—	2	8	no	—
103	C₂²×C₆₇	—	2	2	no	—
104	C₂²×C₆₇	—	2	4	no	—
105	C₂²×C₆₇	—	4	4	no	—
106	C₂²×C₆₇	#519	1	2	yes	—
107	C₂²×C₆₇	C₂×#131	1	4	yes	—
108	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
109	C₄×C₆₇	—	2	2	no	—
110	C₄×C₆₇	—	2	4	no	—
111	C₄×C₆₇	—	2	16	no	—
112	C₄×C₆₇	#519	1	2	yes	—
113	C₄×C₆₇	C₂×#131	1	8	yes	—
114	C₄×C₆₇	C₂×2₋¹⁺⁴	1	2	yes	—
115	C₂×#2986	—	67	1	no	maximal, Sylow
116	C₂×C₆₇	—	4	2	no	—
117	C₂×C₆₇	#2986	1	4	yes	complemented
118	C₂×C₆₇	C₂×#519	1	1	yes	characteristic
119	C₂×C₆₇	C₂²×#131	1	2	yes	—
120	C₂×#533	D₆₇	1	1	yes	characteristic
121	#2986	—	67	16	no	—
122	C₂×#446	—	67	4	no	—
123	C₂×C₈×Q₈	—	67	4	no	—
124	C₂×#433	—	67	4	no	—
125	C₂×#431	—	67	2	no	—
126	C₆₇	C₂×#2986	1	1	yes	characteristic, complemented, Sylow
127	#533	Dic₆₇	1	8	yes	—
128	#533	C₂×D₆₇	1	8	yes	—
129	C₂×C₄⋊Q₈	Dic₆₇	1	2	yes	—
130	C₂×#125	Dic₆₇	1	2	yes	—
131	C₂×C₂²⋊Q₈	Dic₆₇	1	4	yes	—
132	C₂×C₄×Q₈	C₂×D₆₇	1	4	yes	—
133	C₂×#117	C₂×D₆₇	1	2	yes	—
134	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
135	#446	—	67	32	no	—
136	C₈×Q₈	—	67	32	no	—
137	#433	—	67	32	no	—
138	#431	—	67	16	no	—
139	C₂×C₄⋊C₈	—	67	2	no	—
140	C₂×C₄⋊C₈	—	67	2	no	—
141	C₂×C₄⋊C₈	—	67	8	no	—
142	C₂×C₂²⋊C₈	—	67	4	no	—
143	C₂×#97	—	67	4	no	—
144	C₂×C₄×C₈	—	67	4	no	—
145	C₂×C₄×C₈	—	67	4	no	—
146	C₂²×Q₈	C₂×Dic₆₇	1	4	yes	—
147	C₂³×C₄	C₂×Dic₆₇	1	2	yes	—
148	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
149	C₂²×C₈	—	67	4	no	—
150	C₂²×C₈	—	67	4	no	—
151	C₄⋊Q₈	C₂×Dic₆₇	1	16	yes	—
152	#125	C₂×Dic₆₇	1	16	yes	—
153	C₂²⋊Q₈	C₂×Dic₆₇	1	32	yes	—
154	C₄×Q₈	C₂×Dic₆₇	1	16	yes	—
155	C₄×Q₈	C₂²×D₆₇	1	16	yes	—
156	#117	C₂×Dic₆₇	1	8	yes	—
157	#117	C₂²×D₆₇	1	8	yes	—
158	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
159	C₂×C₄⋊C₄	C₂×Dic₆₇	1	8	yes	—
160	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
161	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	4	yes	—
162	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
163	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
164	C₂×C₄²	C₂×Dic₆₇	1	4	yes	—
165	C₂×C₄²	C₂²×D₆₇	1	4	yes	—
166	C₂×C₄²	C₂²×D₆₇	1	4	yes	—
167	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
168	C₄⋊C₈	—	67	8	no	—
169	C₄⋊C₈	—	67	8	no	—
170	C₄⋊C₈	—	67	32	no	—
171	C₂²⋊C₈	—	67	16	no	—
172	#97	—	67	16	no	—
173	C₄×C₈	—	67	16	no	—
174	C₄×C₈	—	67	16	no	—
175	C₂⁴	C₂²×Dic₆₇	1	1	yes	characteristic
176	C₂×Q₈	—	2	16	no	—
177	C₂×Q₈	C₂²×Dic₆₇	1	16	yes	—
178	C₂²×C₄	—	2	8	no	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
181	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
182	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
183	C₂²×C₄	C₂²×Dic₆₇	1	8	yes	—
184	C₂²×C₄	C₂²×Dic₆₇	1	8	yes	—
185	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
186	C₂²×C₄	C₂³×D₆₇	1	2	yes	—
187	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
188	C₂×C₈	—	67	8	no	—
189	C₂×C₈	—	67	8	no	—
190	C₂×C₈	—	134	8	no	—
191	C₂×C₈	—	134	8	no	—
192	C₄⋊C₄	C₂²×Dic₆₇	1	16	yes	—
193	C₄⋊C₄	C₂²×Dic₆₇	1	32	yes	—
194	C₄⋊C₄	C₂²×Dic₆₇	1	16	yes	—
195	C₂²⋊C₄	C₂²×Dic₆₇	1	16	yes	—
196	C₄²	C₂²×Dic₆₇	1	4	yes	—
197	C₄²	C₂²×Dic₆₇	1	4	yes	—
198	C₄²	C₂²×Dic₆₇	1	4	yes	—
199	C₄²	C₂²×Dic₆₇	1	8	yes	—
200	C₄²	C₂³×D₆₇	1	4	yes	—
201	C₄²	C₂³×D₆₇	1	8	yes	—
202	C₂³	—	2	2	no	—
203	C₂³	—	2	4	no	—
204	C₂³	C₂³×Dic₆₇	1	2	yes	—
205	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
206	Q₈	—	2	32	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	16	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	16	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	#14524050	1	8	yes	—
214	C₂×C₄	C₂³×Dic₆₇	1	2	yes	—
215	C₂×C₄	C₂³×Dic₆₇	1	4	yes	—
216	C₂×C₄	C₂³×Dic₆₇	1	16	yes	—
217	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
218	C₂×C₄	C₂³×Dic₆₇	1	4	yes	—
219	C₂×C₄	#14524091	1	4	yes	—
220	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
221	C₈	—	134	8	no	—
222	C₈	—	134	8	no	—
223	C₂²	—	2	4	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	4	4	no	—
226	C₂²	#16863874	1	2	yes	—
227	C₂²	C₂×#14524050	1	4	yes	—
228	C₂²	C₂⁴×Dic₆₇	1	1	yes	characteristic
229	C₄	—	2	4	no	—
230	C₄	—	2	16	no	—
231	C₄	—	2	2	no	—
232	C₄	#16863874	1	2	yes	—
233	C₄	C₂×#14524050	1	8	yes	—
234	C₄	C₂×#14524091	1	2	yes	—
235	C₂	—	4	2	no	—
236	C₂	#22569311	1	4	yes	direct product factor
237	C₂	C₂×#16863874	1	1	yes	characteristic
238	C₂	C₂²×#14524050	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₆₇
commutator	C₂²×C₆₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂×#533×C₆₇	C₂
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂⁴×C₆₇	C₂²×C₄
cosocle	C₂×C₄×C₆₇	C₂⁵
hypercenter	C₂×#533	D₆₇
hypocenter	C₆₇	C₂×#2986
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#533

Complements

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

N	H ≅ G/N	type
1. #22569311	C₂	direct (N × H)
2. C₂×#16863527	C₂	semidirect
3. C₂×Q₈×C₆₇⋊C₈	C₂	semidirect
7. #16863527	C₂²	semidirect
8. Q₈×C₆₇⋊C₈	C₂²	semidirect
117. C₂×C₆₇	#2986	semidirect
126. C₆₇	C₂×#2986	semidirect
236. C₂	#22569311	direct (N × H)

Maximal subgroups

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

subgroup	index
#22569311	2
C₂×#16863527	2
C₂×Q₈×C₆₇⋊C₈	2
C₂×#16863457	2
C₂×#16863449	2
C₂×#533×C₆₇	2
C₂×#2986	67

Subgroups — C2×#22569311 #31720795

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22569311 #31720795