Subgroups — C₂×#22573836 #31762936

Order 17152.

8418 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	#22573836	C₂	1	32	yes	maximal, direct product factor
2	C₂×#16864112	C₂	1	9	yes	maximal, complemented
3	C₂×#16864068	C₂	1	6	yes	maximal, complemented
4	C₂×#16863947	C₂	1	9	yes	maximal, complemented
5	C₂×C₄×#11761629	C₂	1	6	yes	maximal, complemented
6	C₂×Q₈²×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#11761629	C₂²	1	6	yes	—
8	C₂²×#11761629	C₂²	1	6	yes	—
9	#16864112	C₂²	1	144	yes	complemented
10	#16864068	C₂²	1	96	yes	complemented
11	#16863947	C₂²	1	144	yes	complemented
12	C₂×#14524040	C₂²	1	9	yes	—
13	C₂×#14524032	C₂²	1	18	yes	—
14	C₂×#14524018	C₂²	1	36	yes	—
15	C₄×#11761629	C₂²	1	96	yes	complemented
16	C₂×Q₈×Dic₆₇	C₂²	1	2	yes	—
17	C₂×C₄×D₂₆₈	C₂²	1	18	yes	—
18	C₂×#14524005	C₂²	1	18	yes	—
19	C₂×#14524001	C₂²	1	18	yes	—
20	C₂×C₄²×D₆₇	C₂²	1	9	yes	—
21	Q₈²×C₆₇	C₂²	1	16	yes	complemented
22	C₂×C₄⋊Q₈×C₆₇	C₂²	1	9	yes	—
23	C₂×C₄×Q₈×C₆₇	C₂²	1	6	yes	—
24	C₂×#11761629	—	2	48	no	—
25	C₂×#11761629	—	4	24	no	—
26	C₂×#11761629	C₂³	1	48	yes	—
27	C₂×#11761629	C₂³	1	48	yes	—
28	C₂²×D₂₆₈	C₂³	1	18	yes	—
29	C₂²×D₂₆₈	C₂³	1	18	yes	—
30	C₂²×C₄×D₆₇	C₂³	1	18	yes	—
31	C₂²×C₄×D₆₇	C₂³	1	9	yes	—
32	#14524040	C₂³	1	72	yes	—
33	#14524032	C₂³	1	144	yes	—
34	#14524018	C₂³	1	288	yes	—
35	Q₈×Dic₆₇	C₂³	1	16	yes	—
36	C₄×D₂₆₈	C₂³	1	144	yes	—
37	#14524005	C₂³	1	144	yes	—
38	#14524001	C₂³	1	144	yes	—
39	C₂×C₄⋊Dic₆₇	C₂³	1	6	yes	—
40	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	36	yes	—
41	C₂×C₄×Dic₆₇	C₂³	1	6	yes	—
42	C₂×C₄×Dic₆₇	C₂³	1	9	yes	—
43	C₄²×D₆₇	C₂³	1	72	yes	—
44	C₂²×Q₈×C₆₇	C₂³	1	6	yes	—
45	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
46	C₄⋊Q₈×C₆₇	C₂³	1	72	yes	—
47	C₄×Q₈×C₆₇	C₂³	1	48	yes	—
48	C₂×C₄⋊C₄×C₆₇	C₂³	1	18	yes	—
49	C₂×C₄²×C₆₇	C₂³	1	9	yes	—
50	C₂³×D₆₇	C₂⁴	1	9	yes	—
51	#11761629	—	2	192	no	—
52	#11761629	—	4	96	no	—
53	C₂×D₂₆₈	—	2	72	no	—
54	C₂×D₂₆₈	—	4	36	no	—
55	C₂×D₂₆₈	C₂⁴	1	72	yes	—
56	C₂×D₂₆₈	C₂⁴	1	72	yes	—
57	C₂²×Dic₆₇	C₂⁴	1	6	yes	—
58	C₂²×Dic₆₇	C₂⁴	1	1	yes	characteristic
59	C₂×C₄×D₆₇	—	2	72	no	—
60	C₂×C₄×D₆₇	—	2	36	no	—
61	C₂×C₄×D₆₇	C₂⁴	1	72	yes	—
62	C₂×C₄×D₆₇	C₂⁴	1	36	yes	—
63	C₄⋊Dic₆₇	C₂⁴	1	24	yes	—
64	(C₂×D₆₇)⋊C₄	C₂⁴	1	144	yes	—
65	C₄×Dic₆₇	C₂⁴	1	24	yes	—
66	C₄×Dic₆₇	C₂⁴	1	36	yes	—
67	C₂×Q₈×C₆₇	—	4	12	no	—
68	C₂×Q₈×C₆₇	#41	1	16	yes	complemented
69	C₂×Q₈×C₆₇	C₂⁴	1	24	yes	—
70	C₂×Q₈×C₆₇	C₂⁴	1	8	yes	—
71	C₂²×C₄×C₆₇	C₂⁴	1	9	yes	—
72	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
73	C₄⋊C₄×C₆₇	C₂⁴	1	72	yes	—
74	C₄²×C₆₇	C₂⁴	1	36	yes	—
75	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
76	C₂²×D₆₇	—	2	36	no	—
77	C₂²×D₆₇	—	2	18	no	—
78	C₂²×D₆₇	C₂⁵	1	18	yes	—
79	Q₈×C₆₇	—	4	24	no	—
80	Q₈×C₆₇	C₂×#41	1	32	yes	complemented
81	C₂×C₄×C₆₇	—	2	18	no	—
82	C₂×C₄×C₆₇	C₂×#41	1	24	yes	—
83	C₂×C₄×C₆₇	C₂⁵	1	18	yes	—
84	C₂×C₄×C₆₇	C₂⁵	1	12	yes	—
85	C₂×Dic₆₇	—	2	12	no	—
86	C₂×Dic₆₇	H(5,𝔽₂)	1	4	yes	—
87	C₂×Dic₆₇	C₂⁵	1	12	yes	—
88	C₂×Dic₆₇	C₂⁵	1	2	yes	—
89	D₂₆₈	—	2	144	no	—
90	D₂₆₈	—	4	72	no	—
91	C₄×D₆₇	—	2	144	no	—
92	C₄×D₆₇	—	2	72	no	—
93	C₂²×C₆₇	C₂²×#41	1	4	yes	—
94	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
95	C₂²×C₆₇	C₂⁶	1	1	yes	characteristic
96	C₂×D₆₇	—	2	36	no	—
97	C₂×D₆₇	—	2	18	no	—
98	C₂×D₆₇	—	4	36	no	—
99	C₄×C₆₇	—	2	18	no	—
100	C₄×C₆₇	C₂²×#41	1	24	yes	—
101	Dic₆₇	—	2	12	no	—
102	Dic₆₇	C₂×H(5,𝔽₂)	1	4	yes	—
103	C₂×#3483	—	67	1	no	maximal, Sylow
104	C₂×C₆₇	#3483	1	4	yes	complemented
105	C₂×C₆₇	C₂³×#41	1	2	yes	—
106	C₂×C₆₇	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
107	D₆₇	—	4	18	no	—
108	#3483	—	67	32	no	—
109	C₂×Q₈²	D₆₇	1	1	yes	characteristic, complemented
110	C₂×#555	—	67	9	no	—
111	C₂×#550	—	67	6	no	—
112	C₂×#532	—	67	9	no	—
113	C₂×C₄×#41	—	67	6	no	—
114	C₆₇	C₂×#3483	1	1	yes	characteristic, complemented, Sylow
115	C₂²×#41	—	67	6	no	—
116	C₂²×#41	—	67	6	no	—
117	Q₈²	C₂×D₆₇	1	16	yes	complemented
118	#555	—	67	144	no	—
119	#550	—	67	96	no	—
120	#532	—	67	144	no	—
121	C₂×C₄⋊Q₈	C₂×D₆₇	1	9	yes	—
122	C₂×#127	—	67	9	no	—
123	C₂×#124	—	67	18	no	—
124	C₂×#121	—	67	36	no	—
125	C₄×#41	—	67	96	no	—
126	C₂×C₄×Q₈	—	67	2	no	—
127	C₂×C₄×Q₈	C₂×D₆₇	1	6	yes	—
128	C₂×C₄×D₄	—	67	18	no	—
129	C₂×C₄×D₄	—	67	18	no	—
130	C₂×#117	—	67	18	no	—
131	C₂²×C₄²	—	67	9	no	—
132	C₂×#41	—	268	24	no	—
133	C₂×#41	—	67	48	no	—
134	C₂×#41	—	67	48	no	—
135	C₂×#41	—	134	48	no	—
136	C₂²×Q₈	C₂²×D₆₇	1	6	yes	—
137	C₂²×Q₈	C₂²×D₆₇	1	2	yes	—
138	C₂²×D₄	—	67	18	no	—
139	C₂²×D₄	—	67	18	no	—
140	C₂³×C₄	—	67	18	no	—
141	C₂³×C₄	—	67	9	no	—
142	C₄⋊Q₈	C₂²×D₆₇	1	72	yes	—
143	#127	—	67	72	no	—
144	#124	—	67	144	no	—
145	#121	—	67	288	no	—
146	C₄×Q₈	—	67	16	no	—
147	C₄×Q₈	C₂²×D₆₇	1	48	yes	—
148	C₄×D₄	—	67	144	no	—
149	C₄×D₄	—	67	144	no	—
150	#117	—	67	144	no	—
151	C₂×C₄⋊C₄	—	67	6	no	—
152	C₂×C₄⋊C₄	C₂²×D₆₇	1	18	yes	—
153	C₂×C₂²⋊C₄	—	67	36	no	—
154	C₂×C₄²	—	67	9	no	—
155	C₂×C₄²	—	67	6	no	—
156	C₂×C₄²	—	67	72	no	—
157	C₂×C₄²	C₂²×D₆₇	1	9	yes	—
158	C₂⁴	—	67	9	no	—
159	#41	—	268	96	no	—
160	#41	—	134	192	no	—
161	C₂×Q₈	—	4	12	no	—
162	C₂×Q₈	#11761629	1	16	yes	complemented
163	C₂×Q₈	C₂³×D₆₇	1	24	yes	—
164	C₂×Q₈	C₂³×D₆₇	1	8	yes	—
165	C₂×D₄	—	268	36	no	—
166	C₂×D₄	—	67	72	no	—
167	C₂×D₄	—	67	72	no	—
168	C₂×D₄	—	134	72	no	—
169	C₂²×C₄	—	67	1	no	—
170	C₂²×C₄	—	67	6	no	—
171	C₂²×C₄	—	67	72	no	—
172	C₂²×C₄	—	67	36	no	—
173	C₂²×C₄	—	134	36	no	—
174	C₂²×C₄	—	134	72	no	—
175	C₂²×C₄	C₂³×D₆₇	1	9	yes	—
176	C₂²×C₄	C₂³×D₆₇	1	6	yes	—
177	C₄⋊C₄	—	67	24	no	—
178	C₄⋊C₄	C₂³×D₆₇	1	72	yes	—
179	C₂²⋊C₄	—	67	144	no	—
180	C₄²	—	67	24	no	—
181	C₄²	—	67	36	no	—
182	C₄²	C₂³×D₆₇	1	36	yes	—
183	C₂³	—	67	18	no	—
184	C₂³	—	134	36	no	—
185	C₂³	—	134	18	no	—
186	C₂³	C₂⁴×D₆₇	1	1	yes	characteristic
187	Q₈	—	4	24	no	—
188	Q₈	C₂×#11761629	1	32	yes	complemented
189	D₄	—	268	72	no	—
190	D₄	—	134	144	no	—
191	C₂×C₄	—	2	18	no	—
192	C₂×C₄	—	67	4	no	—
193	C₂×C₄	—	67	2	no	—
194	C₂×C₄	—	67	12	no	—
195	C₂×C₄	—	134	72	no	—
196	C₂×C₄	—	134	12	no	—
197	C₂×C₄	—	134	144	no	—
198	C₂×C₄	C₂×#11761629	1	24	yes	—
199	C₂×C₄	C₂⁴×D₆₇	1	18	yes	—
200	C₂×C₄	C₂⁴×D₆₇	1	12	yes	—
201	C₂²	—	268	36	no	—
202	C₂²	—	134	36	no	—
203	C₂²	—	134	18	no	—
204	C₂²	C₂²×#11761629	1	4	yes	—
205	C₂²	H(5,𝔽₂)×D₆₇	1	2	yes	—
206	C₂²	C₂⁵×D₆₇	1	1	yes	characteristic
207	C₄	—	2	18	no	—
208	C₄	—	67	4	no	—
209	C₄	—	134	12	no	—
210	C₄	C₂²×#11761629	1	24	yes	—
211	C₂	—	268	18	no	—
212	C₂	#22573836	1	4	yes	direct product factor
213	C₂	C₂³×#11761629	1	2	yes	—
214	C₂	C₂×H(5,𝔽₂)×D₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂⁴×D₆₇
commutator, cosocle	C₂²×C₆₇	C₂⁶
Frattini	C₂²	C₂⁵×D₆₇
Fitting	C₂×Q₈²×C₆₇	C₂
socle	C₂³×C₆₇	C₂⁵
hypercenter	C₂×Q₈²	D₆₇
hypocenter	C₆₇	C₂×#3483
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×Q₈²

Complements

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

N	H ≅ G/N	type
1. #22573836	C₂	direct (N × H)
2. C₂×#16864112	C₂	semidirect
3. C₂×#16864068	C₂	semidirect
4. C₂×#16863947	C₂	semidirect
5. C₂×C₄×#11761629	C₂	semidirect
6. C₂×Q₈²×C₆₇	C₂	semidirect
9. #16864112	C₂²	semidirect
10. #16864068	C₂²	semidirect
11. #16863947	C₂²	semidirect
15. C₄×#11761629	C₂²	semidirect
21. Q₈²×C₆₇	C₂²	semidirect
68. C₂×Q₈×C₆₇	#41	semidirect
80. Q₈×C₆₇	C₂×#41	semidirect
104. C₂×C₆₇	#3483	semidirect
109. C₂×Q₈²	D₆₇	semidirect
114. C₆₇	C₂×#3483	semidirect
117. Q₈²	C₂×D₆₇	semidirect
162. C₂×Q₈	#11761629	semidirect
188. Q₈	C₂×#11761629	semidirect
212. C₂	#22573836	direct (N × H)

Maximal subgroups

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

subgroup	index
#22573836	2
C₂×#16864112	2
C₂×#16864068	2
C₂×#16863947	2
C₂×C₄×#11761629	2
C₂×Q₈²×C₆₇	2
C₂×#3483	67

Subgroups — C2×#22573836 #31762936

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22573836 #31762936