Subgroups — C₆₇×#2542 #22556973

Order 8576.

676 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊Q₈×C₆₇	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
4	#399×C₆₇	C₂	1	4	yes	maximal, complemented
5	#390×C₆₇	C₂	1	2	yes	maximal
6	#384×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₈×C₆₇	C₂²	1	2	yes	—
9	C₂²×D₄×C₆₇	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₆₇	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₆₇	—	2	4	no	—
12	C₂²⋊Q₈×C₆₇	—	2	16	no	—
13	C₄×D₄×C₆₇	—	2	4	no	—
14	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₆₇	C₂²	1	8	yes	—
16	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	—
21	C₂×C₄²×C₆₇	C₂²	1	2	yes	—
22	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
23	#95×C₆₇	C₂²	1	2	yes	—
24	C₂⁴×C₆₇	C₂³	1	2	yes	—
25	C₂×Q₈×C₆₇	—	2	8	no	—
26	C₂×Q₈×C₆₇	—	2	4	no	—
27	C₂×D₄×C₆₇	—	2	2	no	—
28	C₂×D₄×C₆₇	—	2	2	no	—
29	C₂×D₄×C₆₇	Q₈	1	4	yes	complemented
30	C₂²×C₄×C₆₇	—	2	4	no	—
31	C₂²×C₄×C₆₇	—	2	8	no	—
32	C₂²×C₄×C₆₇	C₂³	1	4	yes	—
33	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
34	C₂²×C₄×C₆₇	C₂³	1	4	yes	—
35	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₆₇	—	2	2	no	—
39	C₄⋊C₄×C₆₇	—	2	16	no	—
40	C₄⋊C₄×C₆₇	—	2	4	no	—
41	C₄⋊C₄×C₆₇	—	2	4	no	—
42	C₄⋊C₄×C₆₇	D₄	1	8	yes	complemented
43	C₂²⋊C₄×C₆₇	—	2	4	no	—
44	C₂²⋊C₄×C₆₇	—	2	16	no	—
45	C₄²×C₆₇	—	2	4	no	—
46	C₄²×C₆₇	D₄	1	4	yes	—
47	C₂³×C₆₇	—	2	4	no	—
48	C₂³×C₆₇	—	2	2	no	—
49	C₂³×C₆₇	—	2	2	no	—
50	C₂³×C₆₇	—	2	4	no	—
51	C₂³×C₆₇	C₂×Q₈	1	4	yes	—
52	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
53	Q₈×C₆₇	—	4	8	no	—
54	D₄×C₆₇	—	2	8	no	—
55	C₂×C₄×C₆₇	—	2	4	no	—
56	C₂×C₄×C₆₇	—	2	2	no	—
57	C₂×C₄×C₆₇	—	2	4	no	—
58	C₂×C₄×C₆₇	—	2	2	no	—
59	C₂×C₄×C₆₇	—	2	8	no	—
60	C₂×C₄×C₆₇	—	2	4	no	—
61	C₂×C₄×C₆₇	—	2	4	no	—
62	C₂×C₄×C₆₇	—	4	8	no	—
63	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
64	C₂×C₄×C₆₇	C₂×D₄	1	8	yes	—
65	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
66	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₆₇	C₂×D₄	1	4	yes	—
68	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
69	C₂×C₄×C₆₇	#41	1	2	yes	—
70	C₂²×C₆₇	—	2	4	no	—
71	C₂²×C₆₇	—	2	8	no	—
72	C₂²×C₆₇	—	4	2	no	—
73	C₂²×C₆₇	—	4	2	no	—
74	C₂²×C₆₇	—	4	4	no	—
75	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
76	C₂²×C₆₇	C₂²×D₄	1	2	yes	—
77	C₂²×C₆₇	C₂²×Q₈	1	1	yes	characteristic
78	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
79	C₂²×C₆₇	2₋¹⁺⁴	1	2	yes	—
80	C₄×C₆₇	—	2	2	no	—
81	C₄×C₆₇	—	2	8	no	—
82	C₄×C₆₇	—	4	2	no	—
83	C₄×C₆₇	—	4	4	no	—
84	C₄×C₆₇	C₂²⋊Q₈	1	4	yes	complemented
85	C₄×C₆₇	#127	1	4	yes	—
86	C₂×C₆₇	—	4	4	no	—
87	C₂×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
88	C₂×C₆₇	C₂×#127	1	1	yes	characteristic
89	C₂×C₆₇	#536	1	1	yes	characteristic
90	C₂×C₆₇	#547	1	2	yes	—
91	C₂×C₆₇	D₄×Q₈	1	2	yes	—
92	#2542	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₆₇	#2542	1	1	yes	characteristic, direct product factor, Sylow
94	C₂×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
95	C₂×C₂²⋊Q₈	C₂×C₆₇	1	4	yes	complemented
96	C₂×C₄×D₄	C₂×C₆₇	1	1	yes	characteristic
97	#399	C₂×C₆₇	1	4	yes	complemented
98	#390	C₂×C₆₇	1	2	yes	—
99	#384	C₂×C₆₇	1	2	yes	complemented
100	C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
101	C₂²×Q₈	C₂²×C₆₇	1	2	yes	—
102	C₂²×D₄	C₂²×C₆₇	1	1	yes	characteristic
103	C₂³×C₄	C₂²×C₆₇	1	2	yes	—
104	C₄⋊Q₈	—	2	4	no	—
105	C₂²⋊Q₈	—	2	16	no	—
106	C₄×D₄	—	2	4	no	—
107	C₂×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
108	C₂×C₄⋊C₄	C₂²×C₆₇	1	8	yes	—
109	C₂×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
110	C₂×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
111	C₂×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
112	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
113	C₂×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	—
114	C₂×C₄²	C₂²×C₆₇	1	2	yes	—
115	C₂×C₄²	C₂²×C₆₇	1	1	yes	characteristic
116	#95	C₂²×C₆₇	1	2	yes	—
117	C₂⁴	C₂³×C₆₇	1	2	yes	—
118	C₂×Q₈	—	2	8	no	—
119	C₂×Q₈	—	2	4	no	—
120	C₂×D₄	—	2	2	no	—
121	C₂×D₄	—	2	2	no	—
122	C₂×D₄	Q₈×C₆₇	1	4	yes	complemented
123	C₂²×C₄	—	2	4	no	—
124	C₂²×C₄	—	2	8	no	—
125	C₂²×C₄	C₂³×C₆₇	1	4	yes	—
126	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
127	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
128	C₂²×C₄	C₂³×C₆₇	1	4	yes	—
129	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
130	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	2	16	no	—
133	C₄⋊C₄	—	2	4	no	—
134	C₄⋊C₄	—	2	4	no	—
135	C₄⋊C₄	D₄×C₆₇	1	8	yes	complemented
136	C₂²⋊C₄	—	2	4	no	—
137	C₂²⋊C₄	—	2	16	no	—
138	C₄²	—	2	4	no	—
139	C₄²	D₄×C₆₇	1	4	yes	—
140	C₂³	—	2	4	no	—
141	C₂³	—	2	2	no	—
142	C₂³	—	2	2	no	—
143	C₂³	—	2	4	no	—
144	C₂³	C₂×Q₈×C₆₇	1	4	yes	—
145	C₂³	C₂⁴×C₆₇	1	1	yes	characteristic
146	Q₈	—	4	8	no	—
147	D₄	—	2	8	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	8	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	4	no	—
155	C₂×C₄	—	4	8	no	—
156	C₂×C₄	C₂×D₄×C₆₇	1	2	yes	—
157	C₂×C₄	C₂×D₄×C₆₇	1	8	yes	—
158	C₂×C₄	C₂×D₄×C₆₇	1	2	yes	—
159	C₂×C₄	C₂×D₄×C₆₇	1	2	yes	—
160	C₂×C₄	C₂×D₄×C₆₇	1	4	yes	—
161	C₂×C₄	C₂×Q₈×C₆₇	1	2	yes	—
162	C₂×C₄	#41×C₆₇	1	2	yes	—
163	C₂²	—	2	4	no	—
164	C₂²	—	2	8	no	—
165	C₂²	—	4	2	no	—
166	C₂²	—	4	2	no	—
167	C₂²	—	4	4	no	—
168	C₂²	C₂²×D₄×C₆₇	1	1	yes	characteristic
169	C₂²	C₂²×D₄×C₆₇	1	2	yes	—
170	C₂²	C₂²×Q₈×C₆₇	1	1	yes	characteristic
171	C₂²	C₂×#41×C₆₇	1	1	yes	characteristic
172	C₂²	2₋¹⁺⁴×C₆₇	1	2	yes	—
173	C₄	—	2	2	no	—
174	C₄	—	2	8	no	—
175	C₄	—	4	2	no	—
176	C₄	—	4	4	no	—
177	C₄	C₂²⋊Q₈×C₆₇	1	4	yes	complemented
178	C₄	#127×C₆₇	1	4	yes	—
179	C₂	—	4	4	no	—
180	C₂	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
181	C₂	C₂×#127×C₆₇	1	1	yes	characteristic
182	C₂	#536×C₆₇	1	1	yes	characteristic
183	C₂	#547×C₆₇	1	2	yes	—
184	C₂	D₄×Q₈×C₆₇	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊Q₈×C₆₇	C₂	semidirect
2. C₂×C₂²⋊Q₈×C₆₇	C₂	semidirect
4. #399×C₆₇	C₂	semidirect
6. #384×C₆₇	C₂	semidirect
7. C₄×C₄⋊C₄×C₆₇	C₂	semidirect
29. C₂×D₄×C₆₇	Q₈	semidirect
42. C₄⋊C₄×C₆₇	D₄	semidirect
84. C₄×C₆₇	C₂²⋊Q₈	semidirect
92. #2542	C₆₇	direct (N × H)
93. C₆₇	#2542	direct (N × H)
94. C₂×C₄⋊Q₈	C₂×C₆₇	semidirect
95. C₂×C₂²⋊Q₈	C₂×C₆₇	semidirect
97. #399	C₂×C₆₇	semidirect
99. #384	C₂×C₆₇	semidirect
100. C₄×C₄⋊C₄	C₂×C₆₇	semidirect
122. C₂×D₄	Q₈×C₆₇	semidirect
135. C₄⋊C₄	D₄×C₆₇	semidirect
177. C₄	C₂²⋊Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Q₈×C₆₇	2
C₂×C₂²⋊Q₈×C₆₇	2
C₂×C₄×D₄×C₆₇	2
#399×C₆₇	2
#390×C₆₇	2
#384×C₆₇	2
C₄×C₄⋊C₄×C₆₇	2
#2542	67

Subgroups — C67×#2542 #22556973

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#2542 #22556973