Subgroups — C₇×Q₈×#101 #13199732

Order 1792.

970 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×Q₈²	C₂	1	1	yes	maximal, characteristic
2	C₇×Q₈×M₁₆	C₂	1	2	yes	maximal
3	C₇×#1997	C₂	1	6	yes	maximal
4	C₇×#1801	C₂	1	3	yes	maximal
5	C₄×C₇×#101	C₂	1	3	yes	maximal
6	C₇×Q₈²	—	2	2	no	—
7	C₇×Q₈²	—	2	6	no	—
8	C₂×C₇×C₄⋊Q₈	C₄	1	6	yes	—
9	C₂×C₇×C₄⋊Q₈	C₂²	1	3	yes	—
10	C₂×C₄×C₇×Q₈	C₄	1	2	yes	—
11	C₂×C₄×C₇×Q₈	C₂²	1	3	yes	—
12	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
13	C₇×#446	—	2	6	no	—
14	C₇×C₈×Q₈	—	2	2	no	—
15	C₇×#423	C₂²	1	6	yes	—
16	C₂×C₇×#101	C₂²	1	3	yes	—
17	C₂×C₇×#101	C₂²	1	1	yes	characteristic
18	C₄×C₇×M₁₆	C₂²	1	6	yes	—
19	C₇×#343	C₂²	1	6	yes	—
20	C₇×#343	C₂²	1	6	yes	—
21	Q₈×#101	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂²×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₇×Q₈	C₂×C₄	1	6	yes	—
24	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
25	C₂×C₇×M₁₆	C₂³	1	6	yes	—
26	C₂×C₇×M₁₆	C₂³	1	2	yes	—
27	C₇×C₄⋊Q₈	—	2	12	no	—
28	C₇×C₄⋊Q₈	—	2	12	no	—
29	C₇×C₄⋊Q₈	—	2	6	no	—
30	C₇×C₄⋊Q₈	—	2	6	no	—
31	C₄×C₇×Q₈	—	2	6	no	—
32	C₄×C₇×Q₈	—	2	2	no	—
33	C₄×C₇×Q₈	—	2	6	no	—
34	C₄×C₇×Q₈	—	2	6	no	—
35	C₄×C₇×Q₈	D₄	1	2	yes	—
36	C₄×C₇×Q₈	D₄	1	6	yes	—
37	C₂×C₇×C₄⋊C₄	C₂×C₄	1	6	yes	—
38	C₂×C₇×C₄⋊C₄	C₂×C₄	1	6	yes	—
39	C₂×C₇×C₄⋊C₄	C₂×C₄	1	3	yes	—
40	C₂×C₇×C₄⋊C₄	C₂³	1	3	yes	—
41	C₂×C₄²×C₇	C₂×C₄	1	6	yes	—
42	C₂×C₄²×C₇	C₂³	1	3	yes	—
43	C₇×C₄⋊C₈	—	2	6	no	—
44	C₇×C₄⋊C₈	—	2	6	no	—
45	C₇×#101	—	2	6	no	—
46	C₇×#101	Q₈	1	4	yes	direct product factor
47	C₇×#97	—	2	6	no	—
48	C₄×C₇×C₈	—	2	6	no	—
49	C₂×Q₈²	C₂×C₇	1	1	yes	characteristic
50	Q₈×M₁₆	C₂×C₇	1	2	yes	—
51	#1997	C₂×C₇	1	6	yes	—
52	#1801	C₂×C₇	1	3	yes	—
53	C₄×#101	C₂×C₇	1	3	yes	—
54	C₂×C₇×Q₈	—	2	4	no	—
55	C₂×C₇×Q₈	—	2	12	no	—
56	C₂×C₇×Q₈	—	2	2	no	—
57	C₂×C₇×Q₈	—	2	2	no	—
58	C₂×C₇×Q₈	—	8	6	no	—
59	C₂×C₇×Q₈	C₂²⋊C₄	1	3	yes	—
60	C₂×C₇×Q₈	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₇×Q₈	C₂×Q₈	1	2	yes	—
62	C₂×C₇×Q₈	#41	1	2	yes	—
63	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
64	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₇	C₂²×C₄	1	6	yes	—
66	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
67	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
68	C₇×M₁₆	—	2	2	no	—
69	C₇×M₁₆	—	2	6	no	—
70	C₇×M₁₆	—	2	6	no	—
71	C₇×M₁₆	C₂×Q₈	1	4	yes	—
72	C₂×C₇×C₈	—	2	2	no	—
73	C₂×C₇×C₈	—	2	6	no	—
74	C₇×C₄⋊C₄	—	2	12	no	—
75	C₇×C₄⋊C₄	—	2	6	no	—
76	C₇×C₄⋊C₄	—	2	6	no	—
77	C₇×C₄⋊C₄	—	2	6	no	—
78	C₇×C₄⋊C₄	C₂²⋊C₄	1	6	yes	—
79	C₇×C₄⋊C₄	C₂²⋊C₄	1	6	yes	—
80	C₄²×C₇	—	2	6	no	—
81	C₄²×C₇	—	2	6	no	—
82	C₄²×C₇	C₂×D₄	1	6	yes	—
83	C₄²×C₇	C₂×D₄	1	6	yes	—
84	Q₈²	—	2	2	no	—
85	Q₈²	—	2	6	no	—
86	C₂×C₄⋊Q₈	C₄×C₇	1	6	yes	—
87	C₂×C₄⋊Q₈	C₂²×C₇	1	3	yes	—
88	C₂×C₄×Q₈	C₄×C₇	1	2	yes	—
89	C₂×C₄×Q₈	C₂²×C₇	1	3	yes	—
90	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
91	#446	—	2	6	no	—
92	C₈×Q₈	—	2	2	no	—
93	#423	C₂²×C₇	1	6	yes	—
94	C₂×#101	C₂²×C₇	1	3	yes	—
95	C₂×#101	C₂²×C₇	1	1	yes	characteristic
96	C₄×M₁₆	C₂²×C₇	1	6	yes	—
97	#343	C₂²×C₇	1	6	yes	—
98	#343	C₂²×C₇	1	6	yes	—
99	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
100	C₇×Q₈	—	2	4	no	—
101	C₇×Q₈	—	2	4	no	—
102	C₇×Q₈	—	2	6	no	—
103	C₇×Q₈	—	8	12	no	—
104	C₇×Q₈	#101	1	4	yes	direct product factor
105	C₂×C₄×C₇	—	2	2	no	—
106	C₂×C₄×C₇	—	2	2	no	—
107	C₂×C₄×C₇	—	2	6	no	—
108	C₂×C₄×C₇	—	2	6	no	—
109	C₂×C₄×C₇	—	4	3	no	—
110	C₂×C₄×C₇	—	4	6	no	—
111	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	3	yes	—
112	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	6	yes	—
113	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	3	yes	—
114	C₂×C₄×C₇	C₄×Q₈	1	4	yes	—
115	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
116	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
117	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
118	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
119	C₇×C₈	—	2	4	no	—
120	C₇×C₈	—	4	6	no	—
121	C₂²×Q₈	C₂×C₄×C₇	1	1	yes	characteristic
122	C₂²×Q₈	C₂×C₄×C₇	1	6	yes	—
123	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
124	C₂×M₁₆	C₂³×C₇	1	6	yes	—
125	C₂×M₁₆	C₂³×C₇	1	2	yes	—
126	C₄⋊Q₈	—	2	12	no	—
127	C₄⋊Q₈	—	2	12	no	—
128	C₄⋊Q₈	—	2	6	no	—
129	C₄⋊Q₈	—	2	6	no	—
130	C₄×Q₈	—	2	6	no	—
131	C₄×Q₈	—	2	2	no	—
132	C₄×Q₈	—	2	6	no	—
133	C₄×Q₈	—	2	6	no	—
134	C₄×Q₈	C₇×D₄	1	2	yes	—
135	C₄×Q₈	C₇×D₄	1	6	yes	—
136	C₂×C₄⋊C₄	C₂×C₄×C₇	1	6	yes	—
137	C₂×C₄⋊C₄	C₂×C₄×C₇	1	6	yes	—
138	C₂×C₄⋊C₄	C₂×C₄×C₇	1	3	yes	—
139	C₂×C₄⋊C₄	C₂³×C₇	1	3	yes	—
140	C₂×C₄²	C₂×C₄×C₇	1	6	yes	—
141	C₂×C₄²	C₂³×C₇	1	3	yes	—
142	C₄⋊C₈	—	2	6	no	—
143	C₄⋊C₈	—	2	6	no	—
144	#101	—	2	6	no	—
145	#101	C₇×Q₈	1	4	yes	direct product factor
146	#97	—	2	6	no	—
147	C₄×C₈	—	2	6	no	—
148	C₂²×C₇	—	2	1	no	—
149	C₂²×C₇	—	2	1	no	—
150	C₂²×C₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
151	C₂²×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
152	C₂²×C₇	#519	1	1	yes	characteristic
153	C₄×C₇	—	2	4	no	—
154	C₄×C₇	—	2	3	no	—
155	C₄×C₇	—	2	6	no	—
156	C₄×C₇	—	4	6	no	—
157	C₄×C₇	C₂×#101	1	6	yes	—
158	C₄×C₇	#548	1	2	yes	—
159	C₄×C₇	D₄×Q₈	1	2	yes	—
160	C₂×Q₈	—	2	4	no	—
161	C₂×Q₈	—	2	12	no	—
162	C₂×Q₈	—	2	2	no	—
163	C₂×Q₈	—	2	2	no	—
164	C₂×Q₈	—	8	6	no	—
165	C₂×Q₈	C₇×C₂²⋊C₄	1	3	yes	—
166	C₂×Q₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
167	C₂×Q₈	C₂×C₇×Q₈	1	2	yes	—
168	C₂×Q₈	C₇×#41	1	2	yes	—
169	C₂²×C₄	C₂²×C₄×C₇	1	3	yes	—
170	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
171	C₂²×C₄	C₂²×C₄×C₇	1	6	yes	—
172	C₂²×C₄	C₂²×C₄×C₇	1	3	yes	—
173	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
174	M₁₆	—	2	2	no	—
175	M₁₆	—	2	6	no	—
176	M₁₆	—	2	6	no	—
177	M₁₆	C₂×C₇×Q₈	1	4	yes	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	6	no	—
180	C₄⋊C₄	—	2	12	no	—
181	C₄⋊C₄	—	2	6	no	—
182	C₄⋊C₄	—	2	6	no	—
183	C₄⋊C₄	—	2	6	no	—
184	C₄⋊C₄	C₇×C₂²⋊C₄	1	6	yes	—
185	C₄⋊C₄	C₇×C₂²⋊C₄	1	6	yes	—
186	C₄²	—	2	6	no	—
187	C₄²	—	2	6	no	—
188	C₄²	C₂×C₇×D₄	1	6	yes	—
189	C₄²	C₂×C₇×D₄	1	6	yes	—
190	C₂×C₇	—	2	1	no	—
191	C₂×C₇	—	2	1	no	—
192	C₂×C₇	Q₈×C₂²⋊C₄	1	1	yes	characteristic
193	C₂×C₇	C₂²×#101	1	1	yes	characteristic
194	C₂×C₇	#2888	1	1	yes	characteristic
195	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
196	Q₈	—	2	4	no	—
197	Q₈	—	2	6	no	—
198	Q₈	—	2	4	no	—
199	Q₈	—	8	12	no	—
200	Q₈	C₇×#101	1	4	yes	direct product factor
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	6	no	—
204	C₂×C₄	—	2	6	no	—
205	C₂×C₄	—	4	3	no	—
206	C₂×C₄	—	4	6	no	—
207	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	3	yes	—
208	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	6	yes	—
209	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	3	yes	—
210	C₂×C₄	C₄×C₇×Q₈	1	4	yes	—
211	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
212	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
213	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
214	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
215	C₈	—	2	4	no	—
216	C₈	—	4	6	no	—
217	C₇	Q₈×#101	1	1	yes	characteristic, direct product factor, Sylow
218	C₂²	—	2	1	no	—
219	C₂²	—	2	1	no	—
220	C₂²	C₂²×C₇×C₂²⋊C₄	1	1	yes	characteristic
221	C₂²	C₂×C₄×C₇×Q₈	1	1	yes	characteristic
222	C₂²	C₇×#519	1	1	yes	characteristic
223	C₄	—	2	4	no	—
224	C₄	—	2	3	no	—
225	C₄	—	2	6	no	—
226	C₄	—	4	6	no	—
227	C₄	C₂×C₇×#101	1	6	yes	—
228	C₄	C₇×#548	1	2	yes	—
229	C₄	C₇×D₄×Q₈	1	2	yes	—
230	C₂	—	2	1	no	—
231	C₂	—	2	1	no	—
232	C₂	C₇×Q₈×C₂²⋊C₄	1	1	yes	characteristic
233	C₂	C₂²×C₇×#101	1	1	yes	characteristic
234	C₂	C₇×#2888	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
21. Q₈×#101	C₇	direct (N × H)
46. C₇×#101	Q₈	direct (N × H)
104. C₇×Q₈	#101	direct (N × H)
145. #101	C₇×Q₈	direct (N × H)
200. Q₈	C₇×#101	direct (N × H)
217. C₇	Q₈×#101	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×Q₈²	2
C₇×Q₈×M₁₆	2
C₇×#1997	2
C₇×#1801	2
C₄×C₇×#101	2
Q₈×#101	7

Subgroups — C7×Q8×#101 #13199732

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×Q₈×#101 #13199732