Subgroups — Q₈×#106×C₆₇ #30701975

Order 17152.

838 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₄⋊C₄×C₆₇	C₂	1	2	yes	maximal
3	C₆₇×#1948	C₂	1	6	yes	maximal
4	C₆₇×#1844	C₂	1	3	yes	maximal
5	C₄×#106×C₆₇	C₂	1	3	yes	maximal
6	C₂×C₄×Q₈×C₆₇	—	2	2	no	—
7	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
8	C₈×Q₈×C₆₇	C₄	1	8	yes	complemented
9	C₂×#106×C₆₇	C₂²	1	3	yes	—
10	C₂×#106×C₆₇	C₂²	1	6	yes	—
11	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈×C₆₇	C₂²	1	3	yes	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	3	yes	—
14	#389×C₆₇	—	2	6	no	—
15	#384×C₆₇	C₂²	1	6	yes	—
16	C₄×C₄⋊C₄×C₆₇	C₂²	1	6	yes	—
17	#340×C₆₇	C₂²	1	6	yes	—
18	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
20	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
21	C₄×Q₈×C₆₇	—	4	8	no	—
22	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
23	C₄×Q₈×C₆₇	Q₈	1	4	yes	—
24	C₂×C₄⋊C₄×C₆₇	—	2	6	no	—
25	C₂×C₄⋊C₄×C₆₇	—	2	6	no	—
26	C₂×C₄⋊C₄×C₆₇	D₄	1	3	yes	—
27	C₂×C₄⋊C₄×C₆₇	C₂³	1	6	yes	—
28	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
29	C₂×C₄²×C₆₇	—	2	6	no	—
30	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
31	#106×C₆₇	—	2	6	no	—
32	#106×C₆₇	—	4	6	no	—
33	#106×C₆₇	Q₈	1	4	yes	direct product factor
34	C₄⋊C₈×C₆₇	C₂×C₄	1	12	yes	—
35	C₄×C₈×C₆₇	C₂×C₄	1	12	yes	—
36	#95×C₆₇	—	2	6	no	—
37	C₂×Q₈×C₆₇	C₄⋊C₄	1	4	yes	—
38	C₂×Q₈×C₆₇	SD₁₆	1	8	yes	—
39	C₂²×C₄×C₆₇	—	2	2	no	—
40	C₂²×C₄×C₆₇	—	2	6	no	—
41	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
42	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
43	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
44	C₂×C₈×C₆₇	—	2	3	no	—
45	C₂×C₈×C₆₇	—	2	3	no	—
46	C₂×C₈×C₆₇	C₂²×C₄	1	6	yes	—
47	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
48	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
49	C₂×C₈×C₆₇	#41	1	2	yes	—
50	C₄⋊C₄×C₆₇	—	2	2	no	—
51	C₄⋊C₄×C₆₇	—	2	6	no	—
52	C₄⋊C₄×C₆₇	—	2	6	no	—
53	C₄⋊C₄×C₆₇	—	4	12	no	—
54	C₄⋊C₄×C₆₇	—	8	6	no	—
55	C₄⋊C₄×C₆₇	C₄⋊C₄	1	12	yes	—
56	C₄⋊C₄×C₆₇	C₂×Q₈	1	4	yes	—
57	C₄²×C₆₇	—	4	12	no	—
58	C₄²×C₆₇	C₂²×C₄	1	6	yes	—
59	C₄²×C₆₇	C₂×Q₈	1	6	yes	—
60	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
61	Q₈×C₆₇	#106	1	16	yes	direct product factor
62	C₂×C₄×C₆₇	—	2	6	no	—
63	C₂×C₄×C₆₇	—	2	4	no	—
64	C₂×C₄×C₆₇	—	4	2	no	—
65	C₂×C₄×C₆₇	—	4	2	no	—
66	C₂×C₄×C₆₇	—	4	6	no	—
67	C₂×C₄×C₆₇	—	4	6	no	—
68	C₂×C₄×C₆₇	—	4	6	no	—
69	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
70	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
71	C₂×C₄×C₆₇	C₂×SD₁₆	1	12	yes	—
72	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
76	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
77	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
78	C₈×C₆₇	—	2	6	no	—
79	C₈×C₆₇	C₄×Q₈	1	4	yes	complemented
80	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₆₇	D₄×Q₈	1	1	yes	characteristic
82	C₂²×C₆₇	#550	1	1	yes	characteristic
83	C₂²×C₆₇	C₂²×SD₁₆	1	2	yes	—
84	C₂²×C₆₇	#577	1	2	yes	—
85	C₄×C₆₇	—	2	6	no	—
86	C₄×C₆₇	—	4	4	no	—
87	C₄×C₆₇	—	8	6	no	—
88	C₄×C₆₇	C₂×#106	1	12	yes	—
89	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
90	C₄×C₆₇	#520	1	1	yes	characteristic
91	C₄×C₆₇	#557	1	1	yes	characteristic
92	C₄×C₆₇	Q₈²	1	1	yes	characteristic
93	Q₈×#106	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂×C₆₇	Q₈×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₆₇	C₂²×#106	1	1	yes	characteristic
96	C₂×C₆₇	#2912	1	1	yes	characteristic
97	C₂×C₆₇	Q₈×SD₁₆	1	2	yes	—
98	C₂×C₆₇	#3392	1	2	yes	—
99	C₂×C₈×Q₈	C₂×C₆₇	1	1	yes	characteristic
100	Q₈×C₄⋊C₄	C₂×C₆₇	1	2	yes	—
101	#1948	C₂×C₆₇	1	6	yes	—
102	#1844	C₂×C₆₇	1	3	yes	—
103	C₄×#106	C₂×C₆₇	1	3	yes	—
104	C₆₇	Q₈×#106	1	1	yes	characteristic, direct product factor, Sylow
105	C₂×C₄×Q₈	—	2	2	no	—
106	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
107	C₈×Q₈	C₄×C₆₇	1	8	yes	complemented
108	C₂×#106	C₂²×C₆₇	1	3	yes	—
109	C₂×#106	C₂²×C₆₇	1	6	yes	—
110	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
111	C₂×C₄⋊C₈	C₂²×C₆₇	1	3	yes	—
112	C₂×C₄×C₈	C₂²×C₆₇	1	3	yes	—
113	#389	—	2	6	no	—
114	#384	C₂²×C₆₇	1	6	yes	—
115	C₄×C₄⋊C₄	C₂²×C₆₇	1	6	yes	—
116	#340	C₂²×C₆₇	1	6	yes	—
117	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
118	C₂²×C₈	C₂³×C₆₇	1	3	yes	—
119	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
120	C₄×Q₈	—	4	8	no	—
121	C₄×Q₈	C₂×C₄×C₆₇	1	4	yes	—
122	C₄×Q₈	Q₈×C₆₇	1	4	yes	—
123	C₂×C₄⋊C₄	—	2	6	no	—
124	C₂×C₄⋊C₄	—	2	6	no	—
125	C₂×C₄⋊C₄	D₄×C₆₇	1	3	yes	—
126	C₂×C₄⋊C₄	C₂³×C₆₇	1	6	yes	—
127	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
128	C₂×C₄²	—	2	6	no	—
129	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
130	#106	—	2	6	no	—
131	#106	—	4	6	no	—
132	#106	Q₈×C₆₇	1	4	yes	direct product factor
133	C₄⋊C₈	C₂×C₄×C₆₇	1	12	yes	—
134	C₄×C₈	C₂×C₄×C₆₇	1	12	yes	—
135	#95	—	2	6	no	—
136	C₂×Q₈	C₄⋊C₄×C₆₇	1	4	yes	—
137	C₂×Q₈	SD₁₆×C₆₇	1	8	yes	—
138	C₂²×C₄	—	2	2	no	—
139	C₂²×C₄	—	2	6	no	—
140	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
141	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
142	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
143	C₂×C₈	—	2	3	no	—
144	C₂×C₈	—	2	3	no	—
145	C₂×C₈	C₂²×C₄×C₆₇	1	6	yes	—
146	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
147	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
148	C₂×C₈	#41×C₆₇	1	2	yes	—
149	C₄⋊C₄	—	2	2	no	—
150	C₄⋊C₄	—	2	6	no	—
151	C₄⋊C₄	—	2	6	no	—
152	C₄⋊C₄	—	4	12	no	—
153	C₄⋊C₄	—	8	6	no	—
154	C₄⋊C₄	C₄⋊C₄×C₆₇	1	12	yes	—
155	C₄⋊C₄	C₂×Q₈×C₆₇	1	4	yes	—
156	C₄²	—	4	12	no	—
157	C₄²	C₂²×C₄×C₆₇	1	6	yes	—
158	C₄²	C₂×Q₈×C₆₇	1	6	yes	—
159	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
160	Q₈	#106×C₆₇	1	16	yes	direct product factor
161	C₂×C₄	—	2	6	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	4	6	no	—
166	C₂×C₄	—	4	6	no	—
167	C₂×C₄	—	4	6	no	—
168	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	6	yes	—
169	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	6	yes	—
170	C₂×C₄	C₂×SD₁₆×C₆₇	1	12	yes	—
171	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
172	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
173	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
174	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
175	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
176	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
177	C₈	—	2	6	no	—
178	C₈	C₄×Q₈×C₆₇	1	4	yes	complemented
179	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
180	C₂²	D₄×Q₈×C₆₇	1	1	yes	characteristic
181	C₂²	#550×C₆₇	1	1	yes	characteristic
182	C₂²	C₂²×SD₁₆×C₆₇	1	2	yes	—
183	C₂²	#577×C₆₇	1	2	yes	—
184	C₄	—	2	6	no	—
185	C₄	—	4	4	no	—
186	C₄	—	8	6	no	—
187	C₄	C₂×#106×C₆₇	1	12	yes	—
188	C₄	C₂×C₄×Q₈×C₆₇	1	1	yes	characteristic
189	C₄	#520×C₆₇	1	1	yes	characteristic
190	C₄	#557×C₆₇	1	1	yes	characteristic
191	C₄	Q₈²×C₆₇	1	1	yes	characteristic
192	C₂	Q₈×C₄⋊C₄×C₆₇	1	1	yes	characteristic
193	C₂	C₂²×#106×C₆₇	1	1	yes	characteristic
194	C₂	C₆₇×#2912	1	1	yes	characteristic
195	C₂	Q₈×SD₁₆×C₆₇	1	2	yes	—
196	C₂	C₆₇×#3392	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄×Q₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₈×Q₈×C₆₇	C₄	semidirect
33. #106×C₆₇	Q₈	direct (N × H)
61. Q₈×C₆₇	#106	direct (N × H)
79. C₈×C₆₇	C₄×Q₈	semidirect
93. Q₈×#106	C₆₇	direct (N × H)
104. C₆₇	Q₈×#106	direct (N × H)
107. C₈×Q₈	C₄×C₆₇	semidirect
132. #106	Q₈×C₆₇	direct (N × H)
160. Q₈	#106×C₆₇	direct (N × H)
178. C₈	C₄×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×C₈×Q₈×C₆₇	2
Q₈×C₄⋊C₄×C₆₇	2
C₆₇×#1948	2
C₆₇×#1844	2
C₄×#106×C₆₇	2
Q₈×#106	67

Subgroups — Q8×#106×C67 #30701975

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#106×C₆₇ #30701975