Subgroups — C₇×#22738 #13199954

Order 1792.

822 conjugacy classes of subgroups.

The subgroup lattice has 186 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×#1834	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	6	yes	—
10	C₂×C₇×C₄⋊C₈	C₂²	1	3	yes	—
11	C₂×C₄×C₇×C₈	C₂²	1	3	yes	—
12	C₇×#389	—	2	6	no	—
13	C₇×#384	C₂²	1	6	yes	—
14	C₄×C₇×C₄⋊C₄	C₂²	1	6	yes	—
15	C₇×#340	C₂²	1	1	yes	characteristic
16	C₇×#340	C₂²	1	3	yes	—
17	C₇×#340	C₂²	1	6	yes	—
18	#22738	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	1	yes	characteristic
21	C₂²×C₇×C₈	C₂³	1	3	yes	—
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	12	no	—
33	C₇×C₄⋊C₈	C₂×C₄	1	12	yes	—
34	C₄×C₇×C₈	C₂×C₄	1	12	yes	—
35	C₇×#95	—	2	6	no	—
36	C₂×C₈×Q₈	C₂×C₇	1	1	yes	characteristic
37	Q₈×C₄⋊C₄	C₂×C₇	1	2	yes	—
38	#1948	C₂×C₇	1	6	yes	—
39	#1834	C₂×C₇	1	3	yes	—
40	C₄×#106	C₂×C₇	1	3	yes	—
41	C₂×C₇×Q₈	C₄⋊C₄	1	4	yes	—
42	C₂×C₇×Q₈	SD₁₆	1	8	yes	—
43	C₂²×C₄×C₇	—	2	2	no	—
44	C₂²×C₄×C₇	—	2	6	no	—
45	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
46	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
47	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
48	C₂×C₇×C₈	—	2	6	no	—
49	C₂×C₇×C₈	—	2	2	no	—
50	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
51	C₂×C₇×C₈	C₂²×C₄	1	6	yes	—
52	C₇×C₄⋊C₄	—	2	2	no	—
53	C₇×C₄⋊C₄	—	2	6	no	—
54	C₇×C₄⋊C₄	—	2	6	no	—
55	C₇×C₄⋊C₄	—	4	12	no	—
56	C₇×C₄⋊C₄	—	8	6	no	—
57	C₇×C₄⋊C₄	C₄⋊C₄	1	12	yes	—
58	C₇×C₄⋊C₄	#41	1	4	yes	—
59	C₄²×C₇	—	4	12	no	—
60	C₄²×C₇	C₂²×C₄	1	6	yes	—
61	C₄²×C₇	C₂×Q₈	1	6	yes	—
62	C₂×C₄×Q₈	—	2	2	no	—
63	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
64	C₈×Q₈	C₄×C₇	1	8	yes	complemented
65	C₂×#106	C₂²×C₇	1	6	yes	—
66	C₂×C₄⋊C₈	C₂²×C₇	1	3	yes	—
67	C₂×C₄×C₈	C₂²×C₇	1	3	yes	—
68	#389	—	2	6	no	—
69	#384	C₂²×C₇	1	6	yes	—
70	C₄×C₄⋊C₄	C₂²×C₇	1	6	yes	—
71	#340	C₂²×C₇	1	3	yes	—
72	#340	C₂²×C₇	1	6	yes	—
73	#340	C₂²×C₇	1	1	yes	characteristic
74	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
75	C₇×Q₈	#106	1	16	yes	complemented
76	C₂×C₄×C₇	—	2	3	no	—
77	C₂×C₄×C₇	—	2	3	no	—
78	C₂×C₄×C₇	—	2	4	no	—
79	C₂×C₄×C₇	—	4	2	no	—
80	C₂×C₄×C₇	—	4	2	no	—
81	C₂×C₄×C₇	—	4	6	no	—
82	C₂×C₄×C₇	—	4	6	no	—
83	C₂×C₄×C₇	—	4	6	no	—
84	C₂×C₄×C₇	C₂×C₄⋊C₄	1	6	yes	—
85	C₂×C₄×C₇	C₂×C₄⋊C₄	1	6	yes	—
86	C₂×C₄×C₇	C₂×SD₁₆	1	12	yes	—
87	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
88	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
89	C₂×C₄×C₇	C₂×#41	1	2	yes	—
90	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
91	C₇×C₈	—	2	6	no	—
92	C₇×C₈	—	2	2	no	—
93	C₂²×Q₈	C₇×D₄	1	1	yes	characteristic
94	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
95	C₂²×C₈	C₂³×C₇	1	3	yes	—
96	C₄×Q₈	—	4	8	no	—
97	C₄×Q₈	C₂×C₄×C₇	1	4	yes	—
98	C₄×Q₈	C₇×Q₈	1	4	yes	—
99	C₂×C₄⋊C₄	—	2	6	no	—
100	C₂×C₄⋊C₄	—	2	6	no	—
101	C₂×C₄⋊C₄	C₇×D₄	1	3	yes	—
102	C₂×C₄⋊C₄	C₂³×C₇	1	6	yes	—
103	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
104	C₂×C₄²	—	2	6	no	—
105	C₂×C₄²	C₂³×C₇	1	3	yes	—
106	#106	—	2	12	no	—
107	C₄⋊C₈	C₂×C₄×C₇	1	12	yes	—
108	C₄×C₈	C₂×C₄×C₇	1	12	yes	—
109	#95	—	2	6	no	—
110	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
111	C₂²×C₇	#548	1	2	yes	—
112	C₂²×C₇	C₂²×SD₁₆	1	2	yes	—
113	C₂²×C₇	#576	1	1	yes	characteristic
114	C₂²×C₇	#578	1	1	yes	characteristic
115	C₄×C₇	—	2	6	no	—
116	C₄×C₇	—	4	4	no	—
117	C₄×C₇	—	8	6	no	—
118	C₄×C₇	C₂×#106	1	12	yes	—
119	C₄×C₇	C₄×#41	1	1	yes	characteristic
120	C₄×C₇	#519	1	1	yes	characteristic
121	C₄×C₇	#557	1	2	yes	—
122	C₂×Q₈	C₇×C₄⋊C₄	1	4	yes	—
123	C₂×Q₈	C₇×SD₁₆	1	8	yes	—
124	C₂²×C₄	—	2	2	no	—
125	C₂²×C₄	—	2	6	no	—
126	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
127	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
128	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
129	C₂×C₈	—	2	6	no	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
132	C₂×C₈	C₂²×C₄×C₇	1	6	yes	—
133	C₄⋊C₄	—	2	2	no	—
134	C₄⋊C₄	—	2	6	no	—
135	C₄⋊C₄	—	2	6	no	—
136	C₄⋊C₄	—	4	12	no	—
137	C₄⋊C₄	—	8	6	no	—
138	C₄⋊C₄	C₇×C₄⋊C₄	1	12	yes	—
139	C₄⋊C₄	C₇×#41	1	4	yes	—
140	C₄²	—	4	12	no	—
141	C₄²	C₂²×C₄×C₇	1	6	yes	—
142	C₄²	C₂×C₇×Q₈	1	6	yes	—
143	C₂×C₇	#2351	1	1	yes	characteristic
144	C₂×C₇	C₂²×#106	1	1	yes	characteristic
145	C₂×C₇	#2913	1	1	yes	characteristic
146	C₂×C₇	#3359	1	2	yes	—
147	C₂×C₇	#3362	1	2	yes	—
148	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
149	Q₈	C₇×#106	1	16	yes	complemented
150	C₂×C₄	—	2	3	no	—
151	C₂×C₄	—	2	3	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	4	6	no	—
156	C₂×C₄	—	4	6	no	—
157	C₂×C₄	—	4	6	no	—
158	C₂×C₄	C₂×C₇×C₄⋊C₄	1	6	yes	—
159	C₂×C₄	C₂×C₇×C₄⋊C₄	1	6	yes	—
160	C₂×C₄	C₂×C₇×SD₁₆	1	12	yes	—
161	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
162	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
163	C₂×C₄	C₂×C₇×#41	1	2	yes	—
164	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
165	C₈	—	2	6	no	—
166	C₈	—	2	2	no	—
167	C₇	#22738	1	1	yes	characteristic, direct product factor, Sylow
168	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
169	C₂²	C₇×#548	1	2	yes	—
170	C₂²	C₂²×C₇×SD₁₆	1	2	yes	—
171	C₂²	C₇×#576	1	1	yes	characteristic
172	C₂²	C₇×#578	1	1	yes	characteristic
173	C₄	—	2	6	no	—
174	C₄	—	4	4	no	—
175	C₄	—	8	6	no	—
176	C₄	C₂×C₇×#106	1	12	yes	—
177	C₄	C₄×C₇×#41	1	1	yes	characteristic
178	C₄	C₇×#519	1	1	yes	characteristic
179	C₄	C₇×#557	1	2	yes	—
180	C₂	C₇×#2351	1	1	yes	characteristic
181	C₂	C₂²×C₇×#106	1	1	yes	characteristic
182	C₂	C₇×#2913	1	1	yes	characteristic
183	C₂	C₇×#3359	1	2	yes	—
184	C₂	C₇×#3362	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. #22738	C₇	direct (N × H)
64. C₈×Q₈	C₄×C₇	semidirect
75. C₇×Q₈	#106	semidirect
149. Q₈	C₇×#106	semidirect
167. C₇	#22738	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×#1834	2
C₄×C₇×#106	2
#22738	7

Subgroups — C7×#22738 #13199954

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22738 #13199954