Subgroups — C₇×Q₈×#108 #13199948

Order 1792.

658 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×Q₈×M₁₆	C₂	1	2	yes	maximal
2	C₂×C₇×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
3	C₇×#1950	C₂	1	6	yes	maximal
4	C₇×#1848	C₂	1	3	yes	maximal
5	C₄×C₇×#108	C₂	1	3	yes	maximal
6	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
7	C₇×#446	—	2	6	no	—
8	C₇×C₈×Q₈	—	2	2	no	—
9	C₇×C₈×Q₈	C₄	1	6	yes	—
10	C₇×C₈×Q₈	C₄	1	2	yes	—
11	C₂×C₇×#108	C₂²	1	3	yes	—
12	C₂×C₇×#108	C₂²	1	6	yes	—
13	C₂×C₇×#108	C₂²	1	1	yes	characteristic
14	C₇×#423	C₂²	1	6	yes	—
15	C₂×C₇×C₄⋊C₈	C₂²	1	3	yes	—
16	C₄×C₇×M₁₆	C₂²	1	6	yes	—
17	C₂×C₄×C₇×C₈	C₂²	1	3	yes	—
18	C₇×#341	C₂²	1	6	yes	—
19	Q₈×#108	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂²×C₇×Q₈	Q₈	1	1	yes	characteristic
21	C₂×C₇×M₁₆	C₂³	1	6	yes	—
22	C₂×C₇×M₁₆	C₂³	1	2	yes	—
23	C₂²×C₇×C₈	C₂³	1	3	yes	—
24	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
25	C₄×C₇×Q₈	C₂×C₄	1	3	yes	—
26	C₄×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
28	C₄×C₇×Q₈	D₄	1	3	yes	—
29	C₂×C₇×C₄⋊C₄	Q₈	1	3	yes	—
30	C₂×C₄²×C₇	C₂³	1	3	yes	—
31	C₇×#108	—	2	6	no	—
32	C₇×#108	—	4	6	no	—
33	C₇×#108	Q₈	1	4	yes	direct product factor
34	C₇×C₄⋊C₈	—	2	6	no	—
35	C₇×C₄⋊C₈	—	2	6	no	—
36	C₇×C₄⋊C₈	C₂×C₄	1	6	yes	—
37	C₇×C₄⋊C₈	C₂×C₄	1	6	yes	—
38	C₇×#97	—	2	6	no	—
39	C₄×C₇×C₈	—	2	6	no	—
40	C₄×C₇×C₈	C₂×C₄	1	6	yes	—
41	C₄×C₇×C₈	C₂×C₄	1	6	yes	—
42	Q₈×M₁₆	C₂×C₇	1	2	yes	—
43	C₂×C₈×Q₈	C₂×C₇	1	1	yes	characteristic
44	#1950	C₂×C₇	1	6	yes	—
45	#1848	C₂×C₇	1	3	yes	—
46	C₄×#108	C₂×C₇	1	3	yes	—
47	C₂×C₇×Q₈	—	2	4	no	—
48	C₂×C₇×Q₈	C₄⋊C₄	1	3	yes	—
49	C₂×C₇×Q₈	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₇	C₂×Q₈	1	3	yes	—
51	C₂²×C₄×C₇	C₂×Q₈	1	3	yes	—
52	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
53	C₇×M₁₆	—	2	2	no	—
54	C₇×M₁₆	—	2	6	no	—
55	C₇×M₁₆	—	2	6	no	—
56	C₇×M₁₆	C₂×Q₈	1	4	yes	—
57	C₂×C₇×C₈	—	2	2	no	—
58	C₂×C₇×C₈	—	2	3	no	—
59	C₂×C₇×C₈	—	2	3	no	—
60	C₂×C₇×C₈	—	2	6	no	—
61	C₂×C₇×C₈	C₂²×C₄	1	6	yes	—
62	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
63	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
64	C₂×C₇×C₈	#41	1	2	yes	—
65	C₇×C₄⋊C₄	C₄⋊C₄	1	6	yes	—
66	C₇×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
67	C₇×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
68	C₄²×C₇	C₂²×C₄	1	3	yes	—
69	C₄²×C₇	C₂²×C₄	1	3	yes	—
70	C₄²×C₇	C₂×D₄	1	3	yes	—
71	C₄²×C₇	C₂×D₄	1	3	yes	—
72	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
73	#446	—	2	6	no	—
74	C₈×Q₈	—	2	2	no	—
75	C₈×Q₈	C₄×C₇	1	6	yes	—
76	C₈×Q₈	C₄×C₇	1	2	yes	—
77	C₂×#108	C₂²×C₇	1	3	yes	—
78	C₂×#108	C₂²×C₇	1	6	yes	—
79	C₂×#108	C₂²×C₇	1	1	yes	characteristic
80	#423	C₂²×C₇	1	6	yes	—
81	C₂×C₄⋊C₈	C₂²×C₇	1	3	yes	—
82	C₄×M₁₆	C₂²×C₇	1	6	yes	—
83	C₂×C₄×C₈	C₂²×C₇	1	3	yes	—
84	#341	C₂²×C₇	1	6	yes	—
85	C₂³×C₇	C₂²×Q₈	1	1	yes	characteristic
86	C₇×Q₈	—	2	6	no	—
87	C₇×Q₈	#108	1	4	yes	direct product factor
88	C₂×C₄×C₇	—	2	6	no	—
89	C₂×C₄×C₇	—	4	3	no	—
90	C₂×C₄×C₇	C₂×C₄⋊C₄	1	3	yes	—
91	C₂×C₄×C₇	C₂×C₄⋊C₄	1	3	yes	—
92	C₂×C₄×C₇	C₂×C₄⋊C₄	1	3	yes	—
93	C₂×C₄×C₇	C₂×C₄⋊C₄	1	3	yes	—
94	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
95	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
97	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
98	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
99	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
100	C₇×C₈	—	2	6	no	—
101	C₇×C₈	—	2	4	no	—
102	C₇×C₈	—	4	6	no	—
103	C₇×C₈	C₄×Q₈	1	4	yes	—
104	C₂²×Q₈	C₇×Q₈	1	1	yes	characteristic
105	C₂×M₁₆	C₂³×C₇	1	6	yes	—
106	C₂×M₁₆	C₂³×C₇	1	2	yes	—
107	C₂²×C₈	C₂³×C₇	1	3	yes	—
108	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
109	C₄×Q₈	C₂×C₄×C₇	1	3	yes	—
110	C₄×Q₈	C₂×C₄×C₇	1	1	yes	characteristic
111	C₄×Q₈	C₇×D₄	1	1	yes	characteristic
112	C₄×Q₈	C₇×D₄	1	3	yes	—
113	C₂×C₄⋊C₄	C₇×Q₈	1	3	yes	—
114	C₂×C₄²	C₂³×C₇	1	3	yes	—
115	#108	—	2	6	no	—
116	#108	—	4	6	no	—
117	#108	C₇×Q₈	1	4	yes	direct product factor
118	C₄⋊C₈	—	2	6	no	—
119	C₄⋊C₈	—	2	6	no	—
120	C₄⋊C₈	C₂×C₄×C₇	1	6	yes	—
121	C₄⋊C₈	C₂×C₄×C₇	1	6	yes	—
122	#97	—	2	6	no	—
123	C₄×C₈	—	2	6	no	—
124	C₄×C₈	C₂×C₄×C₇	1	6	yes	—
125	C₄×C₈	C₂×C₄×C₇	1	6	yes	—
126	C₂²×C₇	—	2	1	no	—
127	C₂²×C₇	—	2	1	no	—
128	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
129	C₂²×C₇	#557	1	1	yes	characteristic
130	C₂²×C₇	Q₈²	1	1	yes	characteristic
131	C₄×C₇	—	2	3	no	—
132	C₄×C₇	—	2	3	no	—
133	C₄×C₇	—	2	3	no	—
134	C₄×C₇	C₂×#108	1	6	yes	—
135	C₄×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
136	C₄×C₇	#520	1	1	yes	characteristic
137	C₄×C₇	D₄×Q₈	1	1	yes	characteristic
138	C₄×C₇	#550	1	1	yes	characteristic
139	C₂×Q₈	—	2	4	no	—
140	C₂×Q₈	C₇×C₄⋊C₄	1	3	yes	—
141	C₂×Q₈	C₇×C₄⋊C₄	1	1	yes	characteristic
142	C₂²×C₄	C₂×C₇×Q₈	1	3	yes	—
143	C₂²×C₄	C₂×C₇×Q₈	1	3	yes	—
144	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
145	M₁₆	—	2	2	no	—
146	M₁₆	—	2	6	no	—
147	M₁₆	—	2	6	no	—
148	M₁₆	C₂×C₇×Q₈	1	4	yes	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	3	no	—
151	C₂×C₈	—	2	3	no	—
152	C₂×C₈	—	2	6	no	—
153	C₂×C₈	C₂²×C₄×C₇	1	6	yes	—
154	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
155	C₂×C₈	C₂×C₇×Q₈	1	2	yes	—
156	C₂×C₈	C₇×#41	1	2	yes	—
157	C₄⋊C₄	C₇×C₄⋊C₄	1	6	yes	—
158	C₄⋊C₄	C₇×C₄⋊C₄	1	3	yes	—
159	C₄⋊C₄	C₇×C₄⋊C₄	1	3	yes	—
160	C₄²	C₂²×C₄×C₇	1	3	yes	—
161	C₄²	C₂²×C₄×C₇	1	3	yes	—
162	C₄²	C₂×C₇×D₄	1	3	yes	—
163	C₄²	C₂×C₇×D₄	1	3	yes	—
164	C₂×C₇	—	2	1	no	—
165	C₂×C₇	—	2	1	no	—
166	C₂×C₇	Q₈×C₄⋊C₄	1	1	yes	characteristic
167	C₂×C₇	C₂²×#108	1	1	yes	characteristic
168	C₂×C₇	#2916	1	1	yes	characteristic
169	C₂³	C₂²×C₇×Q₈	1	1	yes	characteristic
170	Q₈	—	2	6	no	—
171	Q₈	C₇×#108	1	4	yes	direct product factor
172	C₂×C₄	—	2	6	no	—
173	C₂×C₄	—	4	3	no	—
174	C₂×C₄	C₂×C₇×C₄⋊C₄	1	3	yes	—
175	C₂×C₄	C₂×C₇×C₄⋊C₄	1	3	yes	—
176	C₂×C₄	C₂×C₇×C₄⋊C₄	1	3	yes	—
177	C₂×C₄	C₂×C₇×C₄⋊C₄	1	3	yes	—
178	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
179	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
180	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
181	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
182	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
183	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
184	C₈	—	2	6	no	—
185	C₈	—	2	4	no	—
186	C₈	—	4	6	no	—
187	C₈	C₄×C₇×Q₈	1	4	yes	—
188	C₇	Q₈×#108	1	1	yes	characteristic, direct product factor, Sylow
189	C₂²	—	2	1	no	—
190	C₂²	—	2	1	no	—
191	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
192	C₂²	C₇×#557	1	1	yes	characteristic
193	C₂²	C₇×Q₈²	1	1	yes	characteristic
194	C₄	—	2	3	no	—
195	C₄	—	2	3	no	—
196	C₄	—	2	3	no	—
197	C₄	C₂×C₇×#108	1	6	yes	—
198	C₄	C₂×C₄×C₇×Q₈	1	1	yes	characteristic
199	C₄	C₇×#520	1	1	yes	characteristic
200	C₄	C₇×D₄×Q₈	1	1	yes	characteristic
201	C₄	C₇×#550	1	1	yes	characteristic
202	C₂	—	2	1	no	—
203	C₂	—	2	1	no	—
204	C₂	C₇×Q₈×C₄⋊C₄	1	1	yes	characteristic
205	C₂	C₂²×C₇×#108	1	1	yes	characteristic
206	C₂	C₇×#2916	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
19. Q₈×#108	C₇	direct (N × H)
33. C₇×#108	Q₈	direct (N × H)
87. C₇×Q₈	#108	direct (N × H)
117. #108	C₇×Q₈	direct (N × H)
171. Q₈	C₇×#108	direct (N × H)
188. C₇	Q₈×#108	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×Q₈×M₁₆	2
C₂×C₇×C₈×Q₈	2
C₇×#1950	2
C₇×#1848	2
C₄×C₇×#108	2
Q₈×#108	7

Subgroups — C7×Q8×#108 #13199948

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×Q₈×#108 #13199948