Subgroups — C₇×Q₈×#107 #13199942

Order 1792.

838 conjugacy classes of subgroups.

The subgroup lattice has 212 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×#1947	C₂	1	6	yes	maximal
4	C₇×#1845	C₂	1	3	yes	maximal
5	C₄×C₇×#107	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×#107	C₂²	1	3	yes	—
10	C₂×C₇×#107	C₂²	1	6	yes	—
11	C₂×C₇×#107	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×#107	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×#107	—	2	6	no	—
33	C₇×#107	—	4	6	no	—
34	C₇×#107	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	#1947	C₂×C₇	1	6	yes	—
41	#1845	C₂×C₇	1	3	yes	—
42	C₄×#107	C₂×C₇	1	3	yes	—
43	C₂×C₇×Q₈	C₄⋊C₄	1	4	yes	—
44	C₂×C₇×Q₈	D₈	1	4	yes	—
45	C₂×C₇×Q₈	Q₁₆	1	4	yes	—
46	C₂²×C₄×C₇	—	2	2	no	—
47	C₂²×C₄×C₇	—	2	6	no	—
48	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
49	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
50	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
51	C₂×C₇×C₈	—	2	3	no	—
52	C₂×C₇×C₈	—	2	3	no	—
53	C₂×C₇×C₈	C₂²×C₄	1	6	yes	—
54	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
55	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
56	C₂×C₇×C₈	#41	1	2	yes	—
57	C₇×C₄⋊C₄	—	2	2	no	—
58	C₇×C₄⋊C₄	—	2	6	no	—
59	C₇×C₄⋊C₄	—	2	6	no	—
60	C₇×C₄⋊C₄	—	4	12	no	—
61	C₇×C₄⋊C₄	—	8	6	no	—
62	C₇×C₄⋊C₄	C₄⋊C₄	1	12	yes	—
63	C₇×C₄⋊C₄	C₂×Q₈	1	4	yes	—
64	C₄²×C₇	—	4	12	no	—
65	C₄²×C₇	C₂²×C₄	1	6	yes	—
66	C₄²×C₇	C₂×Q₈	1	6	yes	—
67	C₂×C₄×Q₈	—	2	2	no	—
68	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
69	C₈×Q₈	C₄×C₇	1	8	yes	complemented
70	C₂×#107	C₂²×C₇	1	3	yes	—
71	C₂×#107	C₂²×C₇	1	6	yes	—
72	C₂×#107	C₂²×C₇	1	1	yes	characteristic
73	C₂×C₄⋊C₈	C₂²×C₇	1	3	yes	—
74	C₂×C₄×C₈	C₂²×C₇	1	3	yes	—
75	#389	—	2	6	no	—
76	#384	C₂²×C₇	1	6	yes	—
77	C₄×C₄⋊C₄	C₂²×C₇	1	6	yes	—
78	#340	C₂²×C₇	1	6	yes	—
79	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
80	C₇×Q₈	#107	1	16	yes	direct product factor
81	C₂×C₄×C₇	—	2	3	no	—
82	C₂×C₄×C₇	—	2	3	no	—
83	C₂×C₄×C₇	—	2	4	no	—
84	C₂×C₄×C₇	—	4	2	no	—
85	C₂×C₄×C₇	—	4	2	no	—
86	C₂×C₄×C₇	—	4	6	no	—
87	C₂×C₄×C₇	—	4	6	no	—
88	C₂×C₄×C₇	—	4	6	no	—
89	C₂×C₄×C₇	C₂×C₄⋊C₄	1	6	yes	—
90	C₂×C₄×C₇	C₂×C₄⋊C₄	1	6	yes	—
91	C₂×C₄×C₇	C₂×D₈	1	6	yes	—
92	C₂×C₄×C₇	C₂×Q₁₆	1	6	yes	—
93	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
95	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
96	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
97	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
98	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
99	C₇×C₈	—	2	6	no	—
100	C₇×C₈	C₄×Q₈	1	4	yes	complemented
101	C₂²×Q₈	C₇×D₄	1	1	yes	characteristic
102	C₂²×C₈	C₂³×C₇	1	3	yes	—
103	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
104	C₄×Q₈	—	4	8	no	—
105	C₄×Q₈	C₂×C₄×C₇	1	4	yes	—
106	C₄×Q₈	C₇×Q₈	1	4	yes	—
107	C₂×C₄⋊C₄	—	2	6	no	—
108	C₂×C₄⋊C₄	—	2	6	no	—
109	C₂×C₄⋊C₄	C₇×D₄	1	3	yes	—
110	C₂×C₄⋊C₄	C₂³×C₇	1	6	yes	—
111	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
112	C₂×C₄²	—	2	6	no	—
113	C₂×C₄²	C₂³×C₇	1	3	yes	—
114	#107	—	2	6	no	—
115	#107	—	4	6	no	—
116	#107	C₇×Q₈	1	4	yes	direct product factor
117	C₄⋊C₈	C₂×C₄×C₇	1	12	yes	—
118	C₄×C₈	C₂×C₄×C₇	1	12	yes	—
119	#95	—	2	6	no	—
120	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×C₇	D₄×Q₈	1	1	yes	characteristic
122	C₂²×C₇	#550	1	1	yes	characteristic
123	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
124	C₂²×C₇	C₂²×Q₁₆	1	1	yes	characteristic
125	C₂²×C₇	#576	1	1	yes	characteristic
126	C₂²×C₇	#578	1	1	yes	characteristic
127	C₄×C₇	—	2	6	no	—
128	C₄×C₇	—	4	4	no	—
129	C₄×C₇	—	8	6	no	—
130	C₄×C₇	C₂×#107	1	12	yes	—
131	C₄×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
132	C₄×C₇	#520	1	1	yes	characteristic
133	C₄×C₇	#557	1	1	yes	characteristic
134	C₄×C₇	Q₈²	1	1	yes	characteristic
135	C₂×Q₈	C₇×C₄⋊C₄	1	4	yes	—
136	C₂×Q₈	C₇×D₈	1	4	yes	—
137	C₂×Q₈	C₇×Q₁₆	1	4	yes	—
138	C₂²×C₄	—	2	2	no	—
139	C₂²×C₄	—	2	6	no	—
140	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
141	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
142	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
143	C₂×C₈	—	2	3	no	—
144	C₂×C₈	—	2	3	no	—
145	C₂×C₈	C₂²×C₄×C₇	1	6	yes	—
146	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
147	C₂×C₈	C₂×C₇×Q₈	1	2	yes	—
148	C₂×C₈	C₇×#41	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	C₇×C₄⋊C₄	1	12	yes	—
155	C₄⋊C₄	C₂×C₇×Q₈	1	4	yes	—
156	C₄²	—	4	12	no	—
157	C₄²	C₂²×C₄×C₇	1	6	yes	—
158	C₄²	C₂×C₇×Q₈	1	6	yes	—
159	C₂×C₇	Q₈×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₇	C₂²×#107	1	1	yes	characteristic
161	C₂×C₇	#2913	1	1	yes	characteristic
162	C₂×C₇	Q₈×D₈	1	1	yes	characteristic
163	C₂×C₇	Q₈×Q₁₆	1	1	yes	characteristic
164	C₂×C₇	#3391	1	1	yes	characteristic
165	C₂×C₇	#3395	1	1	yes	characteristic
166	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
167	Q₈	C₇×#107	1	16	yes	direct product factor
168	C₂×C₄	—	2	3	no	—
169	C₂×C₄	—	2	3	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	4	6	no	—
174	C₂×C₄	—	4	6	no	—
175	C₂×C₄	—	4	6	no	—
176	C₂×C₄	C₂×C₇×C₄⋊C₄	1	6	yes	—
177	C₂×C₄	C₂×C₇×C₄⋊C₄	1	6	yes	—
178	C₂×C₄	C₂×C₇×D₈	1	6	yes	—
179	C₂×C₄	C₂×C₇×Q₁₆	1	6	yes	—
180	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
181	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
182	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
183	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
184	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
185	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
186	C₈	—	2	6	no	—
187	C₈	C₄×C₇×Q₈	1	4	yes	complemented
188	C₇	Q₈×#107	1	1	yes	characteristic, direct product factor, Sylow
189	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
190	C₂²	C₇×D₄×Q₈	1	1	yes	characteristic
191	C₂²	C₇×#550	1	1	yes	characteristic
192	C₂²	C₂²×C₇×D₈	1	1	yes	characteristic
193	C₂²	C₂²×C₇×Q₁₆	1	1	yes	characteristic
194	C₂²	C₇×#576	1	1	yes	characteristic
195	C₂²	C₇×#578	1	1	yes	characteristic
196	C₄	—	2	6	no	—
197	C₄	—	4	4	no	—
198	C₄	—	8	6	no	—
199	C₄	C₂×C₇×#107	1	12	yes	—
200	C₄	C₂×C₄×C₇×Q₈	1	1	yes	characteristic
201	C₄	C₇×#520	1	1	yes	characteristic
202	C₄	C₇×#557	1	1	yes	characteristic
203	C₄	C₇×Q₈²	1	1	yes	characteristic
204	C₂	C₇×Q₈×C₄⋊C₄	1	1	yes	characteristic
205	C₂	C₂²×C₇×#107	1	1	yes	characteristic
206	C₂	C₇×#2913	1	1	yes	characteristic
207	C₂	C₇×Q₈×D₈	1	1	yes	characteristic
208	C₂	C₇×Q₈×Q₁₆	1	1	yes	characteristic
209	C₂	C₇×#3391	1	1	yes	characteristic
210	C₂	C₇×#3395	1	1	yes	characteristic

	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×#107	C₇	direct (N × H)
34. C₇×#107	Q₈	direct (N × H)
69. C₈×Q₈	C₄×C₇	semidirect
80. C₇×Q₈	#107	direct (N × H)
100. C₇×C₈	C₄×Q₈	semidirect
116. #107	C₇×Q₈	direct (N × H)
167. Q₈	C₇×#107	direct (N × H)
187. C₈	C₄×C₇×Q₈	semidirect
188. C₇	Q₈×#107	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×#1947	2
C₇×#1845	2
C₄×C₇×#107	2
Q₈×#107	7

Subgroups — C7×Q8×#107 #13199942

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×Q₈×#107 #13199942