Subgroups — C₇×Q₈×#106 #13199940

Order 1792.

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×C₇×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₇×Q₈×C₄⋊C₄	C₂	1	2	yes	maximal
3	C₇×#1948	C₂	1	6	yes	maximal
4	C₇×#1844	C₂	1	3	yes	maximal
5	C₄×C₇×#106	C₂	1	3	yes	maximal
6	C₂×C₄×C₇×Q₈	—	2	2	no	—
7	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
8	C₇×C₈×Q₈	C₄	1	8	yes	complemented
9	C₂×C₇×#106	C₂²	1	3	yes	—
10	C₂×C₇×#106	C₂²	1	6	yes	—
11	C₂×C₇×#106	C₂²	1	1	yes	characteristic
12	C₂×C₇×C₄⋊C₈	C₂²	1	3	yes	—
13	C₂×C₄×C₇×C₈	C₂²	1	3	yes	—
14	C₇×#389	—	2	6	no	—
15	C₇×#384	C₂²	1	6	yes	—
16	C₄×C₇×C₄⋊C₄	C₂²	1	6	yes	—
17	C₇×#340	C₂²	1	6	yes	—
18	Q₈×#106	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂²×C₇×Q₈	D₄	1	1	yes	characteristic
20	C₂²×C₇×C₈	C₂³	1	3	yes	—
21	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
22	C₄×C₇×Q₈	—	4	8	no	—
23	C₄×C₇×Q₈	C₂×C₄	1	4	yes	—
24	C₄×C₇×Q₈	Q₈	1	4	yes	—
25	C₂×C₇×C₄⋊C₄	—	2	6	no	—
26	C₂×C₇×C₄⋊C₄	—	2	6	no	—
27	C₂×C₇×C₄⋊C₄	D₄	1	3	yes	—
28	C₂×C₇×C₄⋊C₄	C₂³	1	6	yes	—
29	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
30	C₂×C₄²×C₇	—	2	6	no	—
31	C₂×C₄²×C₇	C₂³	1	3	yes	—
32	C₇×#106	—	2	6	no	—
33	C₇×#106	—	4	6	no	—
34	C₇×#106	Q₈	1	4	yes	direct product factor
35	C₇×C₄⋊C₈	C₂×C₄	1	12	yes	—
36	C₄×C₇×C₈	C₂×C₄	1	12	yes	—
37	C₇×#95	—	2	6	no	—
38	C₂×C₈×Q₈	C₂×C₇	1	1	yes	characteristic
39	Q₈×C₄⋊C₄	C₂×C₇	1	2	yes	—
40	#1948	C₂×C₇	1	6	yes	—
41	#1844	C₂×C₇	1	3	yes	—
42	C₄×#106	C₂×C₇	1	3	yes	—
43	C₂×C₇×Q₈	C₄⋊C₄	1	4	yes	—
44	C₂×C₇×Q₈	SD₁₆	1	8	yes	—
45	C₂²×C₄×C₇	—	2	2	no	—
46	C₂²×C₄×C₇	—	2	6	no	—
47	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
48	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
49	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
50	C₂×C₇×C₈	—	2	3	no	—
51	C₂×C₇×C₈	—	2	3	no	—
52	C₂×C₇×C₈	C₂²×C₄	1	6	yes	—
53	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
54	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
55	C₂×C₇×C₈	#41	1	2	yes	—
56	C₇×C₄⋊C₄	—	2	2	no	—
57	C₇×C₄⋊C₄	—	2	6	no	—
58	C₇×C₄⋊C₄	—	2	6	no	—
59	C₇×C₄⋊C₄	—	4	12	no	—
60	C₇×C₄⋊C₄	—	8	6	no	—
61	C₇×C₄⋊C₄	C₄⋊C₄	1	12	yes	—
62	C₇×C₄⋊C₄	C₂×Q₈	1	4	yes	—
63	C₄²×C₇	—	4	12	no	—
64	C₄²×C₇	C₂²×C₄	1	6	yes	—
65	C₄²×C₇	C₂×Q₈	1	6	yes	—
66	C₂×C₄×Q₈	—	2	2	no	—
67	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
68	C₈×Q₈	C₄×C₇	1	8	yes	complemented
69	C₂×#106	C₂²×C₇	1	3	yes	—
70	C₂×#106	C₂²×C₇	1	6	yes	—
71	C₂×#106	C₂²×C₇	1	1	yes	characteristic
72	C₂×C₄⋊C₈	C₂²×C₇	1	3	yes	—
73	C₂×C₄×C₈	C₂²×C₇	1	3	yes	—
74	#389	—	2	6	no	—
75	#384	C₂²×C₇	1	6	yes	—
76	C₄×C₄⋊C₄	C₂²×C₇	1	6	yes	—
77	#340	C₂²×C₇	1	6	yes	—
78	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
79	C₇×Q₈	#106	1	16	yes	direct product factor
80	C₂×C₄×C₇	—	2	6	no	—
81	C₂×C₄×C₇	—	2	4	no	—
82	C₂×C₄×C₇	—	4	2	no	—
83	C₂×C₄×C₇	—	4	2	no	—
84	C₂×C₄×C₇	—	4	6	no	—
85	C₂×C₄×C₇	—	4	6	no	—
86	C₂×C₄×C₇	—	4	6	no	—
87	C₂×C₄×C₇	C₂×C₄⋊C₄	1	6	yes	—
88	C₂×C₄×C₇	C₂×C₄⋊C₄	1	6	yes	—
89	C₂×C₄×C₇	C₂×SD₁₆	1	12	yes	—
90	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
92	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
93	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
94	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
95	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
96	C₇×C₈	—	2	6	no	—
97	C₇×C₈	C₄×Q₈	1	4	yes	complemented
98	C₂²×Q₈	C₇×D₄	1	1	yes	characteristic
99	C₂²×C₈	C₂³×C₇	1	3	yes	—
100	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
101	C₄×Q₈	—	4	8	no	—
102	C₄×Q₈	C₂×C₄×C₇	1	4	yes	—
103	C₄×Q₈	C₇×Q₈	1	4	yes	—
104	C₂×C₄⋊C₄	—	2	6	no	—
105	C₂×C₄⋊C₄	—	2	6	no	—
106	C₂×C₄⋊C₄	C₇×D₄	1	3	yes	—
107	C₂×C₄⋊C₄	C₂³×C₇	1	6	yes	—
108	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
109	C₂×C₄²	—	2	6	no	—
110	C₂×C₄²	C₂³×C₇	1	3	yes	—
111	#106	—	2	6	no	—
112	#106	—	4	6	no	—
113	#106	C₇×Q₈	1	4	yes	direct product factor
114	C₄⋊C₈	C₂×C₄×C₇	1	12	yes	—
115	C₄×C₈	C₂×C₄×C₇	1	12	yes	—
116	#95	—	2	6	no	—
117	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
118	C₂²×C₇	D₄×Q₈	1	1	yes	characteristic
119	C₂²×C₇	#550	1	1	yes	characteristic
120	C₂²×C₇	C₂²×SD₁₆	1	2	yes	—
121	C₂²×C₇	#577	1	2	yes	—
122	C₄×C₇	—	2	6	no	—
123	C₄×C₇	—	4	4	no	—
124	C₄×C₇	—	8	6	no	—
125	C₄×C₇	C₂×#106	1	12	yes	—
126	C₄×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
127	C₄×C₇	#520	1	1	yes	characteristic
128	C₄×C₇	#557	1	1	yes	characteristic
129	C₄×C₇	Q₈²	1	1	yes	characteristic
130	C₂×Q₈	C₇×C₄⋊C₄	1	4	yes	—
131	C₂×Q₈	C₇×SD₁₆	1	8	yes	—
132	C₂²×C₄	—	2	2	no	—
133	C₂²×C₄	—	2	6	no	—
134	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
135	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
136	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
137	C₂×C₈	—	2	3	no	—
138	C₂×C₈	—	2	3	no	—
139	C₂×C₈	C₂²×C₄×C₇	1	6	yes	—
140	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
141	C₂×C₈	C₂×C₇×Q₈	1	2	yes	—
142	C₂×C₈	C₇×#41	1	2	yes	—
143	C₄⋊C₄	—	2	2	no	—
144	C₄⋊C₄	—	2	6	no	—
145	C₄⋊C₄	—	2	6	no	—
146	C₄⋊C₄	—	4	12	no	—
147	C₄⋊C₄	—	8	6	no	—
148	C₄⋊C₄	C₇×C₄⋊C₄	1	12	yes	—
149	C₄⋊C₄	C₂×C₇×Q₈	1	4	yes	—
150	C₄²	—	4	12	no	—
151	C₄²	C₂²×C₄×C₇	1	6	yes	—
152	C₄²	C₂×C₇×Q₈	1	6	yes	—
153	C₂×C₇	Q₈×C₄⋊C₄	1	1	yes	characteristic
154	C₂×C₇	C₂²×#106	1	1	yes	characteristic
155	C₂×C₇	#2912	1	1	yes	characteristic
156	C₂×C₇	Q₈×SD₁₆	1	2	yes	—
157	C₂×C₇	#3392	1	2	yes	—
158	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
159	Q₈	C₇×#106	1	16	yes	direct product factor
160	C₂×C₄	—	2	6	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	4	6	no	—
165	C₂×C₄	—	4	6	no	—
166	C₂×C₄	—	4	6	no	—
167	C₂×C₄	C₂×C₇×C₄⋊C₄	1	6	yes	—
168	C₂×C₄	C₂×C₇×C₄⋊C₄	1	6	yes	—
169	C₂×C₄	C₂×C₇×SD₁₆	1	12	yes	—
170	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
171	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
172	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
173	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
174	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
175	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
176	C₈	—	2	6	no	—
177	C₈	C₄×C₇×Q₈	1	4	yes	complemented
178	C₇	Q₈×#106	1	1	yes	characteristic, direct product factor, Sylow
179	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
180	C₂²	C₇×D₄×Q₈	1	1	yes	characteristic
181	C₂²	C₇×#550	1	1	yes	characteristic
182	C₂²	C₂²×C₇×SD₁₆	1	2	yes	—
183	C₂²	C₇×#577	1	2	yes	—
184	C₄	—	2	6	no	—
185	C₄	—	4	4	no	—
186	C₄	—	8	6	no	—
187	C₄	C₂×C₇×#106	1	12	yes	—
188	C₄	C₂×C₄×C₇×Q₈	1	1	yes	characteristic
189	C₄	C₇×#520	1	1	yes	characteristic
190	C₄	C₇×#557	1	1	yes	characteristic
191	C₄	C₇×Q₈²	1	1	yes	characteristic
192	C₂	C₇×Q₈×C₄⋊C₄	1	1	yes	characteristic
193	C₂	C₂²×C₇×#106	1	1	yes	characteristic
194	C₂	C₇×#2912	1	1	yes	characteristic
195	C₂	C₇×Q₈×SD₁₆	1	2	yes	—
196	C₂	C₇×#3392	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄×C₇×Q₈ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₇×C₈×Q₈	C₄	semidirect
18. Q₈×#106	C₇	direct (N × H)
34. C₇×#106	Q₈	direct (N × H)
68. C₈×Q₈	C₄×C₇	semidirect
79. C₇×Q₈	#106	direct (N × H)
97. C₇×C₈	C₄×Q₈	semidirect
113. #106	C₇×Q₈	direct (N × H)
159. Q₈	C₇×#106	direct (N × H)
177. C₈	C₄×C₇×Q₈	semidirect
178. C₇	Q₈×#106	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×C₇×C₈×Q₈	2
C₇×Q₈×C₄⋊C₄	2
C₇×#1948	2
C₇×#1844	2
C₄×C₇×#106	2
Q₈×#106	7

Subgroups — C7×Q8×#106 #13199940

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×Q₈×#106 #13199940