Subgroups — Q₈²×Dic₇ #13513802

Order 1792.

2324 conjugacy classes of subgroups.

The subgroup lattice has 176 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	#11740675	C₂	1	9	yes	maximal
3	Dic₇×C₄⋊Q₈	C₂	1	9	yes	maximal
4	Q₈×C₄⋊Dic₇	C₂	1	6	yes	maximal
5	C₄×Q₈×Dic₇	C₂	1	6	yes	maximal
6	C₇×Q₈²	C₄	1	16	yes	complemented
7	C₂×C₇×C₄⋊Q₈	C₂²	1	9	yes	—
8	C₂×C₄×C₇×Q₈	C₂²	1	6	yes	—
9	C₂×Q₈×Dic₇	C₂²	1	6	yes	—
10	C₂×Q₈×Dic₇	C₂²	1	6	yes	—
11	C₂×Q₈×Dic₇	C₂²	1	2	yes	—
12	#90002	C₂²	1	18	yes	—
13	#89757	C₂²	1	36	yes	—
14	C₄⋊C₄×Dic₇	C₂²	1	18	yes	—
15	C₄⋊C₄×Dic₇	C₂²	1	18	yes	—
16	#89707	C₂²	1	9	yes	—
17	C₄×C₄⋊Dic₇	C₂²	1	18	yes	—
18	C₄²×Dic₇	C₂²	1	9	yes	—
19	C₄×Q₈²	—	7	1	no	maximal, Sylow
20	C₂²×C₇×Q₈	C₂³	1	6	yes	—
21	C₂²×C₇×Q₈	C₂³	1	2	yes	—
22	C₇×C₄⋊Q₈	C₂×C₄	1	72	yes	—
23	C₄×C₇×Q₈	C₂×C₄	1	48	yes	—
24	C₂×C₇×C₄⋊C₄	C₂³	1	18	yes	—
25	C₂×C₄²×C₇	C₂³	1	9	yes	—
26	Q₈×Dic₇	—	2	24	no	—
27	Q₈×Dic₇	—	4	12	no	—
28	Q₈×Dic₇	Q₈	1	16	yes	direct product factor
29	C₂×C₄⋊Dic₇	C₂³	1	18	yes	—
30	C₂×C₄⋊Dic₇	C₂³	1	18	yes	—
31	C₂×C₄⋊Dic₇	C₂³	1	6	yes	—
32	C₂×C₄×Dic₇	C₂³	1	18	yes	—
33	C₂×C₄×Dic₇	C₂³	1	9	yes	—
34	C₂×C₄×Dic₇	C₂³	1	9	yes	—
35	C₂×C₄×Dic₇	C₂³	1	6	yes	—
36	C₇⋊#95	C₂³	1	36	yes	—
37	C₂×Q₈²	D₇	1	1	yes	characteristic
38	#2369	—	7	9	no	—
39	Q₈×C₄⋊C₄	—	7	6	no	—
40	C₄×C₄⋊Q₈	—	7	9	no	—
41	C₄²×Q₈	—	7	6	no	—
42	C₂×C₇×Q₈	—	4	6	no	—
43	C₂×C₇×Q₈	—	4	6	no	—
44	C₂×C₇×Q₈	C₂²×C₄	1	24	yes	—
45	C₂×C₇×Q₈	C₂²×C₄	1	8	yes	—
46	C₂×C₇×Q₈	C₂×Q₈	1	8	yes	—
47	C₂×C₇×Q₈	#41	1	8	yes	—
48	C₂²×C₄×C₇	C₂⁴	1	9	yes	—
49	C₂²×C₄×C₇	C₂⁴	1	6	yes	—
50	C₂²×Dic₇	C₂⁴	1	9	yes	—
51	C₂²×Dic₇	C₂⁴	1	6	yes	—
52	C₂²×Dic₇	C₂⁴	1	1	yes	characteristic
53	C₇×C₄⋊C₄	C₂²×C₄	1	72	yes	—
54	C₄²×C₇	C₂²×C₄	1	36	yes	—
55	C₄⋊Dic₇	—	2	36	no	—
56	C₄⋊Dic₇	—	4	18	no	—
57	C₄⋊Dic₇	C₂×Q₈	1	24	yes	—
58	C₄×Dic₇	—	2	36	no	—
59	C₄×Dic₇	—	2	18	no	—
60	C₄×Dic₇	—	4	9	no	—
61	C₄×Dic₇	C₂×Q₈	1	24	yes	—
62	Q₈²	Dic₇	1	16	yes	direct product factor
63	C₂×C₄⋊Q₈	C₂×D₇	1	9	yes	—
64	C₂×C₄×Q₈	—	7	2	no	—
65	C₂×C₄×Q₈	—	7	6	no	—
66	C₂×C₄×Q₈	—	7	6	no	—
67	C₂×C₄×Q₈	C₂×D₇	1	6	yes	—
68	#391	—	7	18	no	—
69	#389	—	7	36	no	—
70	#384	—	7	9	no	—
71	C₄×C₄⋊C₄	—	7	18	no	—
72	C₄×C₄⋊C₄	—	7	18	no	—
73	C₄×C₄⋊C₄	—	7	18	no	—
74	C₄³	—	7	9	no	—
75	C₂³×C₇	C₂⁵	1	1	yes	characteristic
76	C₇×Q₈	—	4	24	no	—
77	C₇×Q₈	C₄×Q₈	1	32	yes	complemented
78	C₂×C₄×C₇	—	2	9	no	—
79	C₂×C₄×C₇	—	2	9	no	—
80	C₂×C₄×C₇	C₂³×C₄	1	18	yes	—
81	C₂×C₄×C₇	C₂³×C₄	1	12	yes	—
82	C₂×C₄×C₇	C₂²×Q₈	1	12	yes	—
83	C₂×C₄×C₇	C₂×#41	1	12	yes	—
84	C₂×Dic₇	—	2	6	no	—
85	C₂×Dic₇	—	2	6	no	—
86	C₂×Dic₇	—	2	18	no	—
87	C₂×Dic₇	—	2	9	no	—
88	C₂×Dic₇	C₂²×Q₈	1	12	yes	—
89	C₂×Dic₇	C₂²×Q₈	1	4	yes	—
90	C₂×Dic₇	H(5,𝔽₂)	1	2	yes	—
91	C₂²×Q₈	C₂²×D₇	1	6	yes	—
92	C₂²×Q₈	C₂²×D₇	1	2	yes	—
93	C₄⋊Q₈	C₂×Dic₇	1	72	yes	—
94	C₄×Q₈	—	7	16	no	—
95	C₄×Q₈	—	14	24	no	—
96	C₄×Q₈	—	28	12	no	—
97	C₄×Q₈	C₂×Dic₇	1	48	yes	—
98	C₂×C₄⋊C₄	—	7	6	no	—
99	C₂×C₄⋊C₄	—	7	18	no	—
100	C₂×C₄⋊C₄	—	7	18	no	—
101	C₂×C₄⋊C₄	C₂²×D₇	1	18	yes	—
102	C₂×C₄²	—	7	6	no	—
103	C₂×C₄²	—	7	9	no	—
104	C₂×C₄²	—	7	18	no	—
105	C₂×C₄²	—	7	9	no	—
106	C₂×C₄²	C₂²×D₇	1	9	yes	—
107	#95	—	7	36	no	—
108	C₂²×C₇	C₂⁴×C₄	1	1	yes	characteristic
109	C₂²×C₇	C₂³×Q₈	1	2	yes	—
110	C₂²×C₇	C₂²×#41	1	2	yes	—
111	C₂²×C₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
112	C₂²×C₇	#585	1	1	yes	characteristic
113	C₄×C₇	—	2	18	no	—
114	C₄×C₇	C₂×C₄×Q₈	1	24	yes	—
115	Dic₇	—	2	12	no	—
116	Dic₇	—	4	9	no	—
117	Dic₇	Q₈²	1	4	yes	direct product factor
118	C₂×Q₈	—	4	6	no	—
119	C₂×Q₈	—	4	6	no	—
120	C₂×Q₈	Q₈×D₇	1	8	yes	—
121	C₂×Q₈	#1161	1	8	yes	—
122	C₂×Q₈	C₂²×Dic₇	1	24	yes	—
123	C₂×Q₈	C₂²×Dic₇	1	8	yes	—
124	C₂²×C₄	—	7	1	no	—
125	C₂²×C₄	—	7	6	no	—
126	C₂²×C₄	—	7	9	no	—
127	C₂²×C₄	C₂³×D₇	1	9	yes	—
128	C₂²×C₄	C₂³×D₇	1	6	yes	—
129	C₄⋊C₄	—	7	24	no	—
130	C₄⋊C₄	—	14	36	no	—
131	C₄⋊C₄	—	28	18	no	—
132	C₄⋊C₄	C₂²×Dic₇	1	72	yes	—
133	C₄²	—	7	24	no	—
134	C₄²	—	14	18	no	—
135	C₄²	—	14	36	no	—
136	C₄²	—	28	9	no	—
137	C₄²	C₂²×Dic₇	1	36	yes	—
138	C₂×C₇	C₂²×C₄×Q₈	1	2	yes	—
139	C₂×C₇	C₄×H(5,𝔽₂)	1	1	yes	characteristic
140	C₂×C₇	C₂×Q₈²	1	1	yes	characteristic
141	C₂×C₇	Q₈×#41	1	2	yes	—
142	C₂×C₇	#3483	1	1	yes	characteristic
143	C₂³	C₂⁴×D₇	1	1	yes	characteristic
144	Q₈	—	4	24	no	—
145	Q₈	Q₈×Dic₇	1	32	yes	direct product factor
146	C₂×C₄	—	2	9	no	—
147	C₂×C₄	—	2	9	no	—
148	C₂×C₄	—	7	4	no	—
149	C₂×C₄	—	7	2	no	—
150	C₂×C₄	—	7	12	no	—
151	C₂×C₄	—	14	6	no	—
152	C₂×C₄	—	14	6	no	—
153	C₂×C₄	—	14	18	no	—
154	C₂×C₄	—	14	9	no	—
155	C₂×C₄	C₂×Q₈×D₇	1	12	yes	—
156	C₂×C₄	C₂×#1161	1	12	yes	—
157	C₂×C₄	C₂³×Dic₇	1	18	yes	—
158	C₂×C₄	C₂³×Dic₇	1	12	yes	—
159	C₇	C₄×Q₈²	1	1	yes	characteristic, complemented, Sylow
160	C₂²	C₂²×Q₈×D₇	1	2	yes	—
161	C₂²	C₂²×#1161	1	2	yes	—
162	C₂²	D₇×H(5,𝔽₂)	1	1	yes	characteristic
163	C₂²	#90618	1	1	yes	characteristic
164	C₂²	C₂⁴×Dic₇	1	1	yes	characteristic
165	C₄	—	2	18	no	—
166	C₄	—	7	4	no	—
167	C₄	—	14	12	no	—
168	C₄	—	28	9	no	—
169	C₄	C₂×Q₈×Dic₇	1	24	yes	—
170	C₂	Q₈×#1161	1	2	yes	—
171	C₂	#11744108	1	1	yes	characteristic
172	C₂	Q₈²×D₇	1	1	yes	characteristic
173	C₂	C₂²×Q₈×Dic₇	1	2	yes	—
174	C₂	Dic₇×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₇
commutator	C₂²×C₇	C₂⁴×C₄
Fitting	C₂×C₇×Q₈²	C₂
socle, prime radical, cosocle	C₂³×C₇	C₂⁵
hypercenter	C₂×Q₈²	D₇
hypocenter	C₇	C₄×Q₈²
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×Q₈²

Complements

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

N	H ≅ G/N	type
6. C₇×Q₈²	C₄	semidirect
28. Q₈×Dic₇	Q₈	direct (N × H)
62. Q₈²	Dic₇	direct (N × H)
77. C₇×Q₈	C₄×Q₈	semidirect
117. Dic₇	Q₈²	direct (N × H)
145. Q₈	Q₈×Dic₇	direct (N × H)
159. C₇	C₄×Q₈²	semidirect

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
#11740675	2
Dic₇×C₄⋊Q₈	2
Q₈×C₄⋊Dic₇	2
C₄×Q₈×Dic₇	2
C₄×Q₈²	7

Subgroups — Q82×Dic7 #13513802

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈²×Dic₇ #13513802