Subgroups — C₂×#22573215 #31736760

Order 17152.

1894 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16864125	C₂	1	2	yes	maximal
2	#22573215	C₂	1	16	yes	maximal, direct product factor
3	C₂×#16863776	C₂	1	3	yes	maximal
4	C₂×#16863607	C₂	1	6	yes	maximal
5	C₂×C₄×Dic₂₆₈	C₂	1	3	yes	maximal
6	C₂×C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₆₈	C₂²	1	3	yes	—
8	C₂²×Dic₂₆₈	C₂²	1	6	yes	—
9	#16864125	—	2	8	no	—
10	#16864125	C₂²	1	16	yes	—
11	C₂×#14524043	C₂²	1	6	yes	—
12	C₂×#14524034	—	2	6	no	—
13	C₂×Q₈×Dic₆₇	—	2	2	no	—
14	C₂×C₄×Dic₁₃₄	C₂²	1	6	yes	—
15	#16863776	C₂²	1	24	yes	—
16	#16863607	C₂²	1	48	yes	—
17	C₄×Dic₂₆₈	C₂²	1	24	yes	—
18	C₂×#14523977	C₂²	1	1	yes	characteristic
19	C₂×#14523968	C₂²	1	6	yes	—
20	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₈×Q₈×C₆₇	C₂²	1	8	yes	—
22	C₂×C₄⋊C₈×C₆₇	C₂²	1	3	yes	—
23	C₂×C₄×C₈×C₆₇	C₂²	1	3	yes	—
24	C₂²×Dic₁₃₄	C₂³	1	6	yes	—
25	C₂×Dic₂₆₈	—	2	12	no	—
26	C₂×Dic₂₆₈	—	4	12	no	—
27	C₂×Dic₂₆₈	C₂³	1	12	yes	—
28	C₂×Dic₂₆₈	C₂³	1	24	yes	—
29	#14524043	—	2	12	no	—
30	#14524043	C₂³	1	24	yes	—
31	#14524034	—	2	48	no	—
32	Q₈×Dic₆₇	—	2	16	no	—
33	C₄×Dic₁₃₄	—	2	12	no	—
34	C₄×Dic₁₃₄	C₂³	1	24	yes	—
35	C₂×C₄⋊Dic₆₇	—	2	6	no	—
36	C₂×C₄⋊Dic₆₇	—	2	6	no	—
37	C₂×C₄⋊Dic₆₇	C₂³	1	2	yes	—
38	C₂×Dic₆₇⋊C₄	—	2	6	no	—
39	C₂×C₄×Dic₆₇	—	2	6	no	—
40	#14523977	C₂³	1	4	yes	—
41	#14523968	C₂³	1	24	yes	—
42	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
44	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
45	C₄×Q₈×C₆₇	D₄	1	4	yes	—
46	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
47	C₂×C₄⋊C₄×C₆₇	D₄	1	3	yes	—
48	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
49	C₄⋊C₈×C₆₇	C₂³	1	12	yes	—
50	C₄×C₈×C₆₇	C₂³	1	12	yes	—
51	C₂×Dic₁₃₄	—	2	12	no	—
52	C₂×Dic₁₃₄	—	2	12	no	—
53	C₂×Dic₁₃₄	—	2	6	no	—
54	C₂×Dic₁₃₄	C₂⁴	1	12	yes	—
55	C₂²×Dic₆₇	—	2	2	no	—
56	C₂²×Dic₆₇	—	2	6	no	—
57	Dic₂₆₈	—	2	24	no	—
58	Dic₂₆₈	—	4	24	no	—
59	C₄⋊Dic₆₇	—	2	24	no	—
60	C₄⋊Dic₆₇	—	2	24	no	—
61	C₄⋊Dic₆₇	—	2	2	no	—
62	C₄⋊Dic₆₇	C₂⁴	1	4	yes	—
63	Dic₆₇⋊C₄	—	2	24	no	—
64	C₄×Dic₆₇	—	2	24	no	—
65	C₂×Q₈×C₆₇	Q₁₆	1	8	yes	complemented
66	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
67	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
68	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
69	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
70	C₂×C₈×C₆₇	—	2	6	no	—
71	C₂×C₈×C₆₇	#41	1	4	yes	—
72	C₂×C₈×C₆₇	C₂⁴	1	6	yes	—
73	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
74	C₄⋊C₄×C₆₇	C₂×D₄	1	12	yes	—
75	C₄²×C₆₇	C₂×D₄	1	6	yes	—
76	C₄²×C₆₇	C₂⁴	1	6	yes	—
77	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
78	Q₈×C₆₇	C₂×Q₁₆	1	16	yes	complemented
79	C₂×C₄×C₆₇	—	2	6	no	—
80	C₂×C₄×C₆₇	C₂×Q₁₆	1	12	yes	—
81	C₂×C₄×C₆₇	C₂²×D₄	1	6	yes	—
82	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
83	C₂×C₄×C₆₇	C₂²×D₄	1	6	yes	—
84	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
85	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
86	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
87	C₂×Dic₆₇	—	2	4	no	—
88	C₂×Dic₆₇	—	2	12	no	—
89	C₂×Dic₆₇	—	4	12	no	—
90	C₂×Dic₆₇	—	4	4	no	—
91	Dic₁₃₄	—	2	12	no	—
92	Dic₁₃₄	—	2	12	no	—
93	Dic₁₃₄	—	4	12	no	—
94	C₈×C₆₇	—	2	6	no	—
95	C₈×C₆₇	C₂×#41	1	4	yes	—
96	C₂²×C₆₇	#550	1	2	yes	—
97	C₂²×C₆₇	C₂²×Q₁₆	1	2	yes	—
98	C₂²×C₆₇	#578	1	2	yes	—
99	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
100	C₄×C₆₇	—	2	6	no	—
101	C₄×C₆₇	#550	1	2	yes	—
102	C₄×C₆₇	C₂²×Q₁₆	1	12	yes	—
103	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
104	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
105	Dic₆₇	—	4	12	no	—
106	Dic₆₇	—	4	4	no	—
107	C₂×#3395	—	67	1	no	maximal, Sylow
108	C₂×C₆₇	#3395	1	4	yes	complemented
109	C₂×C₆₇	C₂×#550	1	1	yes	characteristic
110	C₂×C₆₇	C₂³×Q₁₆	1	1	yes	characteristic
111	C₂×C₆₇	C₂×#578	1	1	yes	characteristic
112	C₂×#557	—	67	2	no	—
113	#3395	—	67	16	no	—
114	C₂×#494	—	67	3	no	—
115	C₂×#462	—	67	6	no	—
116	C₂×C₈×Q₈	D₆₇	1	1	yes	characteristic
117	C₂×C₄×Q₁₆	—	67	3	no	—
118	C₆₇	C₂×#3395	1	1	yes	characteristic, complemented, Sylow
119	C₂²×Q₁₆	—	67	6	no	—
120	C₂²×Q₁₆	—	67	3	no	—
121	#557	—	67	16	no	—
122	#557	—	134	8	no	—
123	C₂×C₄⋊Q₈	—	67	6	no	—
124	C₂×#125	—	134	6	no	—
125	C₂×C₄×Q₈	—	67	6	no	—
126	C₂×C₄×Q₈	—	134	2	no	—
127	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
128	#494	—	67	24	no	—
129	#462	—	67	48	no	—
130	C₈×Q₈	C₂×D₆₇	1	8	yes	—
131	C₄×Q₁₆	—	67	24	no	—
132	C₂×#107	—	67	1	no	—
133	C₂×C₄⋊C₈	C₂×D₆₇	1	3	yes	—
134	C₂×#103	—	67	6	no	—
135	C₂×C₄×C₈	C₂×D₆₇	1	3	yes	—
136	C₂²×Q₈	—	67	6	no	—
137	C₂²×Q₈	D₂₆₈	1	1	yes	characteristic
138	C₂×Q₁₆	—	268	12	no	—
139	C₂×Q₁₆	—	67	24	no	—
140	C₂×Q₁₆	—	67	12	no	—
141	C₂×Q₁₆	—	134	12	no	—
142	C₂²×C₈	C₂²×D₆₇	1	3	yes	—
143	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
144	C₄⋊Q₈	—	67	24	no	—
145	C₄⋊Q₈	—	134	12	no	—
146	#125	—	134	48	no	—
147	C₄×Q₈	—	67	24	no	—
148	C₄×Q₈	—	134	16	no	—
149	C₄×Q₈	—	134	12	no	—
150	C₄×Q₈	D₂₆₈	1	4	yes	—
151	C₄×Q₈	C₂²×D₆₇	1	4	yes	—
152	C₂×C₄⋊C₄	—	67	2	no	—
153	C₂×C₄⋊C₄	—	134	6	no	—
154	C₂×C₄⋊C₄	—	134	6	no	—
155	C₂×C₄⋊C₄	—	134	6	no	—
156	C₂×C₄⋊C₄	D₂₆₈	1	3	yes	—
157	C₂×C₄²	—	134	6	no	—
158	C₂×C₄²	C₂²×D₆₇	1	3	yes	—
159	#107	—	67	4	no	—
160	C₄⋊C₈	C₂²×D₆₇	1	12	yes	—
161	#103	—	67	24	no	—
162	C₄×C₈	C₂²×D₆₇	1	12	yes	—
163	C₂×Q₈	—	67	12	no	—
164	C₂×Q₈	—	134	12	no	—
165	C₂×Q₈	—	134	12	no	—
166	C₂×Q₈	—	134	6	no	—
167	C₂×Q₈	Dic₂₆₈	1	8	yes	complemented
168	C₂×Q₈	C₂×D₂₆₈	1	4	yes	—
169	C₂²×C₄	—	134	2	no	—
170	C₂²×C₄	—	134	6	no	—
171	C₂²×C₄	C₂×D₂₆₈	1	3	yes	—
172	C₂²×C₄	C₂×D₂₆₈	1	3	yes	—
173	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
174	Q₁₆	—	268	24	no	—
175	Q₁₆	—	134	24	no	—
176	C₂×C₈	—	2	6	no	—
177	C₂×C₈	#11761629	1	4	yes	—
178	C₂×C₈	C₂³×D₆₇	1	6	yes	—
179	C₂×C₈	C₂³×D₆₇	1	2	yes	—
180	C₄⋊C₄	—	67	4	no	—
181	C₄⋊C₄	—	134	2	no	—
182	C₄⋊C₄	—	134	24	no	—
183	C₄⋊C₄	—	134	24	no	—
184	C₄⋊C₄	—	134	24	no	—
185	C₄⋊C₄	C₂×D₂₆₈	1	12	yes	—
186	C₄²	—	134	24	no	—
187	C₄²	C₂×D₂₆₈	1	6	yes	—
188	C₄²	C₂³×D₆₇	1	6	yes	—
189	C₂³	C₂²×D₂₆₈	1	1	yes	characteristic
190	Q₈	—	268	12	no	—
191	Q₈	—	134	12	no	—
192	Q₈	—	134	12	no	—
193	Q₈	C₂×Dic₂₆₈	1	16	yes	complemented
194	C₂×C₄	—	2	6	no	—
195	C₂×C₄	—	268	12	no	—
196	C₂×C₄	—	268	4	no	—
197	C₂×C₄	—	134	4	no	—
198	C₂×C₄	—	134	12	no	—
199	C₂×C₄	C₂×Dic₂₆₈	1	12	yes	—
200	C₂×C₄	C₂²×D₂₆₈	1	6	yes	—
201	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
202	C₂×C₄	C₂²×D₂₆₈	1	6	yes	—
203	C₂×C₄	C₂×#11761629	1	2	yes	—
204	C₂×C₄	#14524089	1	2	yes	—
205	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
206	C₈	—	2	6	no	—
207	C₈	C₂×#11761629	1	4	yes	—
208	C₂²	#16864068	1	2	yes	—
209	C₂²	C₂²×Dic₂₆₈	1	2	yes	—
210	C₂²	#16864213	1	2	yes	—
211	C₂²	C₂³×D₂₆₈	1	1	yes	characteristic
212	C₄	—	2	6	no	—
213	C₄	—	268	12	no	—
214	C₄	—	268	4	no	—
215	C₄	#16864068	1	2	yes	—
216	C₄	C₂²×Dic₂₆₈	1	12	yes	—
217	C₄	C₂²×#11761629	1	1	yes	characteristic
218	C₄	C₂×#14524089	1	1	yes	characteristic
219	C₂	#22573215	1	4	yes	direct product factor
220	C₂	C₂×#16864068	1	1	yes	characteristic
221	C₂	C₂³×Dic₂₆₈	1	1	yes	characteristic
222	C₂	C₂×#16864213	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₂₆₈
commutator, cosocle	C₂×C₄×C₆₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂×C₈×Q₈×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂²×D₄
hypercenter	C₂×C₈×Q₈	D₆₇
hypocenter	C₆₇	C₂×#3395
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. #22573215	C₂	direct (N × H)
65. C₂×Q₈×C₆₇	Q₁₆	semidirect
78. Q₈×C₆₇	C₂×Q₁₆	semidirect
108. C₂×C₆₇	#3395	semidirect
118. C₆₇	C₂×#3395	semidirect
167. C₂×Q₈	Dic₂₆₈	semidirect
193. Q₈	C₂×Dic₂₆₈	semidirect
219. C₂	#22573215	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×#16864125	2
#22573215	2
C₂×#16863776	2
C₂×#16863607	2
C₂×C₄×Dic₂₆₈	2
C₂×C₈×Q₈×C₆₇	2
C₂×#3395	67

Subgroups — C2×#22573215 #31736760

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22573215 #31736760