Subgroups — C₂×#554×C₆₇ #22557903

Order 8576.

1200 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	#554×C₆₇	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#125×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₂×C₂²⋊Q₈×C₆₇	C₂	1	4	yes	maximal, complemented
5	C₂×C₂²⋊Q₈×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×D₄×C₆₇	C₂	1	2	yes	maximal
8	C₂×C₄×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₄⋊C₄×C₆₇	C₂	1	2	yes	maximal, complemented
10	C₂²×Q₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂²×Q₈×C₆₇	C₂²	1	2	yes	—
12	C₂²×D₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₆₇	C₂²	1	2	yes	—
14	C₂³×C₄×C₆₇	C₂²	1	4	yes	—
15	C₄⋊Q₈×C₆₇	C₂²	1	8	yes	complemented
16	#125×C₆₇	C₂²	1	16	yes	complemented
17	C₂²⋊Q₈×C₆₇	C₂²	1	32	yes	complemented
18	C₂²⋊Q₈×C₆₇	C₂²	1	16	yes	complemented
19	C₄×Q₈×C₆₇	C₂²	1	8	yes	complemented
20	C₄×D₄×C₆₇	C₂²	1	16	yes	—
21	C₄×D₄×C₆₇	C₂²	1	8	yes	—
22	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
24	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
25	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
27	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
28	C₂×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
29	C₂×C₄⋊C₄×C₆₇	C₂²	1	16	yes	complemented
30	C₂×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
31	C₂×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
32	C₂×C₄²×C₆₇	C₂²	1	2	yes	—
33	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
34	C₂⁴×C₆₇	C₂³	1	2	yes	—
35	C₂×Q₈×C₆₇	—	2	4	no	—
36	C₂×Q₈×C₆₇	—	4	4	no	—
37	C₂×Q₈×C₆₇	C₂³	1	4	yes	—
38	C₂×Q₈×C₆₇	C₂³	1	8	yes	—
39	C₂×D₄×C₆₇	Q₈	1	8	yes	complemented
40	C₂×D₄×C₆₇	C₂³	1	4	yes	—
41	C₂²×C₄×C₆₇	—	2	8	no	—
42	C₂²×C₄×C₆₇	—	4	8	no	—
43	C₂²×C₄×C₆₇	C₂³	1	8	yes	—
44	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
45	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
47	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
48	C₂²×C₄×C₆₇	C₂³	1	4	yes	—
49	C₂²×C₄×C₆₇	C₂³	1	16	yes	—
50	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
52	C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
53	C₄⋊C₄×C₆₇	C₂³	1	16	yes	—
54	C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
55	C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
56	C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
57	C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
58	C₄⋊C₄×C₆₇	C₂³	1	16	yes	—
59	C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
60	C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
61	C₄²×C₆₇	C₂³	1	8	yes	—
62	C₄²×C₆₇	C₂³	1	4	yes	—
63	C₂³×C₆₇	—	2	4	no	—
64	C₂³×C₆₇	—	2	4	no	—
65	C₂³×C₆₇	C₂×Q₈	1	8	yes	—
66	C₂³×C₆₇	C₂⁴	1	4	yes	—
67	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
68	Q₈×C₆₇	—	2	8	no	—
69	Q₈×C₆₇	—	4	8	no	—
70	D₄×C₆₇	C₂×Q₈	1	16	yes	complemented
71	C₂×C₄×C₆₇	—	2	4	no	—
72	C₂×C₄×C₆₇	—	2	4	no	—
73	C₂×C₄×C₆₇	—	2	4	no	—
74	C₂×C₄×C₆₇	—	2	8	no	—
75	C₂×C₄×C₆₇	—	2	2	no	—
76	C₂×C₄×C₆₇	—	2	16	no	—
77	C₂×C₄×C₆₇	—	4	16	no	—
78	C₂×C₄×C₆₇	C₂×Q₈	1	4	yes	—
79	C₂×C₄×C₆₇	#41	1	4	yes	—
80	C₂×C₄×C₆₇	C₂⁴	1	4	yes	—
81	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
82	C₂×C₄×C₆₇	C₂⁴	1	4	yes	—
83	C₂×C₄×C₆₇	C₂⁴	1	4	yes	—
84	C₂×C₄×C₆₇	C₂⁴	1	8	yes	—
85	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
86	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
87	C₂²×C₆₇	—	2	16	no	—
88	C₂²×C₆₇	—	2	4	no	—
89	C₂²×C₆₇	—	2	4	no	—
90	C₂²×C₆₇	C₂²×Q₈	1	8	yes	—
91	C₂²×C₆₇	C₂²×Q₈	1	2	yes	—
92	C₂²×C₆₇	C₂×#41	1	2	yes	—
93	C₂²×C₆₇	H(5,𝔽₂)	1	2	yes	—
94	C₂²×C₆₇	C₂⁵	1	1	yes	characteristic
95	C₄×C₆₇	—	2	4	no	—
96	C₄×C₆₇	—	2	4	no	—
97	C₄×C₆₇	—	2	4	no	—
98	C₄×C₆₇	—	2	8	no	—
99	C₄×C₆₇	—	2	2	no	—
100	C₄×C₆₇	C₂²×Q₈	1	4	yes	—
101	C₄×C₆₇	C₂×#41	1	4	yes	—
102	C₂×C₆₇	—	2	8	no	—
103	C₂×C₆₇	#554	1	4	yes	direct product factor
104	C₂×C₆₇	C₂³×Q₈	1	1	yes	characteristic
105	C₂×C₆₇	C₂²×#41	1	1	yes	characteristic
106	C₂×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
107	C₂×#554	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₆₇	C₂×#554	1	1	yes	characteristic, direct product factor, Sylow
109	#554	C₂×C₆₇	1	16	yes	direct product factor
110	C₂×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
111	C₂×#125	C₂×C₆₇	1	2	yes	complemented
112	C₂×C₂²⋊Q₈	C₂×C₆₇	1	4	yes	complemented
113	C₂×C₂²⋊Q₈	C₂×C₆₇	1	2	yes	complemented
114	C₂×C₄×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
115	C₂×C₄×D₄	C₂×C₆₇	1	2	yes	—
116	C₂×C₄×D₄	C₂×C₆₇	1	1	yes	characteristic
117	C₂²×C₄⋊C₄	C₂×C₆₇	1	2	yes	complemented
118	C₂²×Q₈	C₂²×C₆₇	1	1	yes	characteristic
119	C₂²×Q₈	C₂²×C₆₇	1	2	yes	—
120	C₂²×D₄	C₂²×C₆₇	1	1	yes	characteristic
121	C₂³×C₄	C₂²×C₆₇	1	2	yes	—
122	C₂³×C₄	C₂²×C₆₇	1	4	yes	—
123	C₄⋊Q₈	C₂²×C₆₇	1	8	yes	complemented
124	#125	C₂²×C₆₇	1	16	yes	complemented
125	C₂²⋊Q₈	C₂²×C₆₇	1	32	yes	complemented
126	C₂²⋊Q₈	C₂²×C₆₇	1	16	yes	complemented
127	C₄×Q₈	C₂²×C₆₇	1	8	yes	complemented
128	C₄×D₄	C₂²×C₆₇	1	16	yes	—
129	C₄×D₄	C₂²×C₆₇	1	8	yes	—
130	C₂×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
131	C₂×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
132	C₂×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
133	C₂×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
134	C₂×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
135	C₂×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
136	C₂×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
137	C₂×C₄⋊C₄	C₂²×C₆₇	1	16	yes	complemented
138	C₂×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
139	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
140	C₂×C₄²	C₂²×C₆₇	1	2	yes	—
141	C₂×C₄²	C₂²×C₆₇	1	1	yes	characteristic
142	C₂⁴	C₂³×C₆₇	1	2	yes	—
143	C₂×Q₈	—	2	4	no	—
144	C₂×Q₈	—	4	4	no	—
145	C₂×Q₈	C₂³×C₆₇	1	4	yes	—
146	C₂×Q₈	C₂³×C₆₇	1	8	yes	—
147	C₂×D₄	Q₈×C₆₇	1	8	yes	complemented
148	C₂×D₄	C₂³×C₆₇	1	4	yes	—
149	C₂²×C₄	—	2	8	no	—
150	C₂²×C₄	—	4	8	no	—
151	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
152	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
153	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
154	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
155	C₂²×C₄	C₂³×C₆₇	1	4	yes	—
156	C₂²×C₄	C₂³×C₆₇	1	8	yes	—
157	C₂²×C₄	C₂³×C₆₇	1	16	yes	—
158	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
159	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
160	C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
161	C₄⋊C₄	C₂³×C₆₇	1	16	yes	—
162	C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
163	C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
164	C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
165	C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
166	C₄⋊C₄	C₂³×C₆₇	1	16	yes	—
167	C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
168	C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
169	C₄²	C₂³×C₆₇	1	8	yes	—
170	C₄²	C₂³×C₆₇	1	4	yes	—
171	C₂³	—	2	4	no	—
172	C₂³	—	2	4	no	—
173	C₂³	C₂×Q₈×C₆₇	1	8	yes	—
174	C₂³	C₂⁴×C₆₇	1	4	yes	—
175	C₂³	C₂⁴×C₆₇	1	1	yes	characteristic
176	Q₈	—	2	8	no	—
177	Q₈	—	4	8	no	—
178	D₄	C₂×Q₈×C₆₇	1	16	yes	complemented
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	2	8	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	16	no	—
185	C₂×C₄	—	4	16	no	—
186	C₂×C₄	C₂×Q₈×C₆₇	1	4	yes	—
187	C₂×C₄	#41×C₆₇	1	4	yes	—
188	C₂×C₄	C₂⁴×C₆₇	1	4	yes	—
189	C₂×C₄	C₂⁴×C₆₇	1	2	yes	—
190	C₂×C₄	C₂⁴×C₆₇	1	4	yes	—
191	C₂×C₄	C₂⁴×C₆₇	1	4	yes	—
192	C₂×C₄	C₂⁴×C₆₇	1	8	yes	—
193	C₂×C₄	C₂⁴×C₆₇	1	2	yes	—
194	C₂×C₄	C₂⁴×C₆₇	1	2	yes	—
195	C₂²	—	2	16	no	—
196	C₂²	—	2	4	no	—
197	C₂²	—	2	4	no	—
198	C₂²	C₂²×Q₈×C₆₇	1	8	yes	—
199	C₂²	C₂²×Q₈×C₆₇	1	2	yes	—
200	C₂²	C₂×#41×C₆₇	1	2	yes	—
201	C₂²	H(5,𝔽₂)×C₆₇	1	2	yes	—
202	C₂²	C₂⁵×C₆₇	1	1	yes	characteristic
203	C₄	—	2	4	no	—
204	C₄	—	2	4	no	—
205	C₄	—	2	4	no	—
206	C₄	—	2	8	no	—
207	C₄	—	2	2	no	—
208	C₄	C₂²×Q₈×C₆₇	1	4	yes	—
209	C₄	C₂×#41×C₆₇	1	4	yes	—
210	C₂	—	2	8	no	—
211	C₂	#554×C₆₇	1	4	yes	direct product factor
212	C₂	C₂³×Q₈×C₆₇	1	1	yes	characteristic
213	C₂	C₂²×#41×C₆₇	1	1	yes	characteristic
214	C₂	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₆₇
prime radical	C₂²×D₄×C₆₇	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. #554×C₆₇	C₂	direct (N × H)
2. C₂×C₄⋊Q₈×C₆₇	C₂	semidirect
3. C₂×#125×C₆₇	C₂	semidirect
4. C₂×C₂²⋊Q₈×C₆₇	C₂	semidirect
5. C₂×C₂²⋊Q₈×C₆₇	C₂	semidirect
6. C₂×C₄×Q₈×C₆₇	C₂	semidirect
9. C₂²×C₄⋊C₄×C₆₇	C₂	semidirect
15. C₄⋊Q₈×C₆₇	C₂²	semidirect
16. #125×C₆₇	C₂²	semidirect
17. C₂²⋊Q₈×C₆₇	C₂²	semidirect
18. C₂²⋊Q₈×C₆₇	C₂²	semidirect
19. C₄×Q₈×C₆₇	C₂²	semidirect
29. C₂×C₄⋊C₄×C₆₇	C₂²	semidirect
39. C₂×D₄×C₆₇	Q₈	semidirect
70. D₄×C₆₇	C₂×Q₈	semidirect
103. C₂×C₆₇	#554	direct (N × H)
107. C₂×#554	C₆₇	direct (N × H)
108. C₆₇	C₂×#554	direct (N × H)
109. #554	C₂×C₆₇	direct (N × H)
110. C₂×C₄⋊Q₈	C₂×C₆₇	semidirect
111. C₂×#125	C₂×C₆₇	semidirect
112. C₂×C₂²⋊Q₈	C₂×C₆₇	semidirect
113. C₂×C₂²⋊Q₈	C₂×C₆₇	semidirect
114. C₂×C₄×Q₈	C₂×C₆₇	semidirect
117. C₂²×C₄⋊C₄	C₂×C₆₇	semidirect
123. C₄⋊Q₈	C₂²×C₆₇	semidirect
124. #125	C₂²×C₆₇	semidirect
125. C₂²⋊Q₈	C₂²×C₆₇	semidirect
126. C₂²⋊Q₈	C₂²×C₆₇	semidirect
127. C₄×Q₈	C₂²×C₆₇	semidirect
137. C₂×C₄⋊C₄	C₂²×C₆₇	semidirect
147. C₂×D₄	Q₈×C₆₇	semidirect
178. D₄	C₂×Q₈×C₆₇	semidirect
211. C₂	#554×C₆₇	direct (N × H)

Maximal subgroups

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

subgroup	index
#554×C₆₇	2
C₂×C₄⋊Q₈×C₆₇	2
C₂×#125×C₆₇	2
C₂×C₂²⋊Q₈×C₆₇	2
C₂×C₄×Q₈×C₆₇	2
C₂×C₄×D₄×C₆₇	2
C₂²×C₄⋊C₄×C₆₇	2
C₂×#554	67

Subgroups — C2×#554×C67 #22557903

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#554×C₆₇ #22557903