Subgroups — C₇×#36608 #13213824

Order 1792.

1626 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2862	C₂	1	3	yes	maximal
2	C₇×#2581	C₂	1	6	yes	maximal
3	C₇×#2514	C₂	1	12	yes	maximal
4	C₇×#2356	C₂	1	6	yes	maximal
5	C₄×C₇×C₄⋊Q₈	C₂	1	3	yes	maximal
6	C₄²×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
7	C₂×C₇×C₄⋊Q₈	C₂²	1	3	yes	—
8	C₂×C₄×C₇×Q₈	C₂²	1	3	yes	—
9	C₂×C₄×C₇×Q₈	C₂²	1	6	yes	—
10	C₇×#401	C₂²	1	8	yes	—
11	C₇×#400	C₂²	1	12	yes	—
12	C₇×#395	C₂²	1	4	yes	—
13	C₇×#391	C₂²	1	6	yes	—
14	C₇×#391	C₂²	1	12	yes	—
15	C₇×#391	C₂²	1	12	yes	—
16	C₇×#389	C₂²	1	12	yes	—
17	C₇×#387	C₂²	1	24	yes	—
18	C₇×#384	C₂²	1	1	yes	characteristic
19	C₇×#383	C₂²	1	12	yes	—
20	C₇×#382	C₂²	1	6	yes	—
21	C₇×#382	C₂²	1	6	yes	—
22	C₄×C₇×C₄⋊C₄	C₂²	1	6	yes	—
23	C₄×C₇×C₄⋊C₄	C₂²	1	12	yes	—
24	C₄×C₇×C₄⋊C₄	C₂²	1	3	yes	—
25	C₇×#376	C₂²	1	4	yes	—
26	C₄³×C₇	C₂²	1	3	yes	—
27	#36608	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
29	C₂²×C₇×Q₈	C₂³	1	6	yes	—
30	C₇×C₄⋊Q₈	—	2	12	no	—
31	C₄×C₇×Q₈	—	2	12	no	—
32	C₄×C₇×Q₈	—	2	24	no	—
33	C₂×C₇×C₄⋊C₄	C₂³	1	6	yes	—
34	C₂×C₇×C₄⋊C₄	C₂³	1	6	yes	—
35	C₂×C₇×C₄⋊C₄	C₂³	1	3	yes	—
36	C₂×C₇×C₄⋊C₄	C₂³	1	12	yes	—
37	C₂×C₇×C₄⋊C₄	C₂³	1	12	yes	—
38	C₂×C₄²×C₇	C₂³	1	6	yes	—
39	C₂×C₄²×C₇	C₂³	1	3	yes	—
40	C₂×C₄²×C₇	C₂³	1	6	yes	—
41	C₂×C₄²×C₇	C₂³	1	12	yes	—
42	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₇	C₂³	1	3	yes	—
44	C₇×#95	C₂³	1	1	yes	characteristic
45	C₇×#95	C₂³	1	12	yes	—
46	C₇×#95	C₂³	1	3	yes	—
47	C₇×#95	C₂³	1	2	yes	—
48	C₇×#95	C₂³	1	24	yes	—
49	C₇×#95	C₂³	1	24	yes	—
50	C₇×#95	C₂³	1	12	yes	—
51	#2862	C₂×C₇	1	3	yes	—
52	#2581	C₂×C₇	1	6	yes	—
53	#2514	C₂×C₇	1	12	yes	—
54	#2356	C₂×C₇	1	6	yes	—
55	C₄×C₄⋊Q₈	C₂×C₇	1	3	yes	—
56	C₄²×Q₈	C₂×C₇	1	1	yes	characteristic
57	C₂×C₇×Q₈	—	2	6	no	—
58	C₂×C₇×Q₈	—	2	12	no	—
59	C₂×C₇×Q₈	—	2	12	no	—
60	C₂×C₇×Q₈	—	2	12	no	—
61	C₂²×C₄×C₇	C₂⁴	1	6	yes	—
62	C₂²×C₄×C₇	C₂⁴	1	12	yes	—
63	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
64	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
65	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
66	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
67	C₇×C₄⋊C₄	—	2	12	no	—
68	C₇×C₄⋊C₄	—	2	6	no	—
69	C₇×C₄⋊C₄	—	2	12	no	—
70	C₇×C₄⋊C₄	—	2	24	no	—
71	C₇×C₄⋊C₄	—	2	24	no	—
72	C₄²×C₇	—	2	12	no	—
73	C₄²×C₇	—	2	24	no	—
74	C₄²×C₇	—	2	12	no	—
75	C₄²×C₇	—	2	6	no	—
76	C₄²×C₇	#41	1	12	yes	—
77	C₄²×C₇	#41	1	4	yes	—
78	C₂×C₄⋊Q₈	C₂²×C₇	1	3	yes	—
79	C₂×C₄×Q₈	C₂²×C₇	1	6	yes	—
80	C₂×C₄×Q₈	C₂²×C₇	1	3	yes	—
81	#401	C₂²×C₇	1	8	yes	—
82	#400	C₂²×C₇	1	12	yes	—
83	#395	C₂²×C₇	1	4	yes	—
84	#391	C₂²×C₇	1	6	yes	—
85	#391	C₂²×C₇	1	12	yes	—
86	#391	C₂²×C₇	1	12	yes	—
87	#389	C₂²×C₇	1	12	yes	—
88	#387	C₂²×C₇	1	24	yes	—
89	#384	C₂²×C₇	1	1	yes	characteristic
90	#383	C₂²×C₇	1	12	yes	—
91	#382	C₂²×C₇	1	6	yes	—
92	#382	C₂²×C₇	1	6	yes	—
93	C₄×C₄⋊C₄	C₂²×C₇	1	12	yes	—
94	C₄×C₄⋊C₄	C₂²×C₇	1	3	yes	—
95	C₄×C₄⋊C₄	C₂²×C₇	1	6	yes	—
96	#376	C₂²×C₇	1	4	yes	—
97	C₄³	C₂²×C₇	1	3	yes	—
98	C₂³×C₇	C₂⁵	1	1	yes	characteristic
99	C₇×Q₈	—	2	8	no	—
100	C₇×Q₈	—	4	24	no	—
101	C₂×C₄×C₇	—	2	6	no	—
102	C₂×C₄×C₇	—	2	6	no	—
103	C₂×C₄×C₇	—	2	12	no	—
104	C₂×C₄×C₇	—	2	6	no	—
105	C₂×C₄×C₇	—	2	12	no	—
106	C₂×C₄×C₇	—	2	12	no	—
107	C₂×C₄×C₇	—	2	6	no	—
108	C₂×C₄×C₇	—	2	6	no	—
109	C₂×C₄×C₇	—	2	12	no	—
110	C₂×C₄×C₇	C₂×#41	1	6	yes	—
111	C₂×C₄×C₇	C₂×#41	1	6	yes	—
112	C₂×C₄×C₇	C₂×#41	1	6	yes	—
113	C₂×C₄×C₇	C₂×#41	1	6	yes	—
114	C₂×C₄×C₇	2₋¹⁺⁴	1	6	yes	—
115	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
116	C₂²×Q₈	C₂³×C₇	1	6	yes	—
117	C₄⋊Q₈	—	2	12	no	—
118	C₄×Q₈	—	2	12	no	—
119	C₄×Q₈	—	2	24	no	—
120	C₂×C₄⋊C₄	C₂³×C₇	1	6	yes	—
121	C₂×C₄⋊C₄	C₂³×C₇	1	6	yes	—
122	C₂×C₄⋊C₄	C₂³×C₇	1	3	yes	—
123	C₂×C₄⋊C₄	C₂³×C₇	1	12	yes	—
124	C₂×C₄⋊C₄	C₂³×C₇	1	12	yes	—
125	C₂×C₄²	C₂³×C₇	1	6	yes	—
126	C₂×C₄²	C₂³×C₇	1	12	yes	—
127	C₂×C₄²	C₂³×C₇	1	6	yes	—
128	C₂×C₄²	C₂³×C₇	1	3	yes	—
129	C₂×C₄²	C₂³×C₇	1	3	yes	—
130	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
131	#95	C₂³×C₇	1	1	yes	characteristic
132	#95	C₂³×C₇	1	12	yes	—
133	#95	C₂³×C₇	1	3	yes	—
134	#95	C₂³×C₇	1	2	yes	—
135	#95	C₂³×C₇	1	24	yes	—
136	#95	C₂³×C₇	1	24	yes	—
137	#95	C₂³×C₇	1	12	yes	—
138	C₂²×C₇	C₂²×#41	1	3	yes	—
139	C₂²×C₇	C₂²×#41	1	1	yes	characteristic
140	C₂²×C₇	C₂×2₋¹⁺⁴	1	3	yes	—
141	C₄×C₇	—	2	6	no	—
142	C₄×C₇	—	2	6	no	—
143	C₄×C₇	—	4	4	no	—
144	C₄×C₇	—	4	12	no	—
145	C₄×C₇	—	4	6	no	—
146	C₄×C₇	#556	1	12	yes	—
147	C₂×Q₈	—	2	6	no	—
148	C₂×Q₈	—	2	12	no	—
149	C₂×Q₈	—	2	12	no	—
150	C₂×Q₈	—	2	12	no	—
151	C₂²×C₄	C₂⁴×C₇	1	6	yes	—
152	C₂²×C₄	C₂⁴×C₇	1	12	yes	—
153	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
154	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
155	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
156	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
157	C₄⋊C₄	—	2	12	no	—
158	C₄⋊C₄	—	2	6	no	—
159	C₄⋊C₄	—	2	12	no	—
160	C₄⋊C₄	—	2	24	no	—
161	C₄⋊C₄	—	2	24	no	—
162	C₄²	—	2	12	no	—
163	C₄²	—	2	24	no	—
164	C₄²	—	2	12	no	—
165	C₄²	—	2	6	no	—
166	C₄²	C₇×#41	1	12	yes	—
167	C₄²	C₇×#41	1	4	yes	—
168	C₂×C₇	#3444	1	1	yes	characteristic
169	C₂×C₇	C₂×#556	1	3	yes	—
170	C₂×C₇	#3509	1	3	yes	—
171	C₂³	C₂⁵×C₇	1	1	yes	characteristic
172	Q₈	—	2	8	no	—
173	Q₈	—	4	24	no	—
174	C₂×C₄	—	2	6	no	—
175	C₂×C₄	—	2	6	no	—
176	C₂×C₄	—	2	12	no	—
177	C₂×C₄	—	2	6	no	—
178	C₂×C₄	—	2	12	no	—
179	C₂×C₄	—	2	12	no	—
180	C₂×C₄	—	2	6	no	—
181	C₂×C₄	—	2	6	no	—
182	C₂×C₄	—	2	12	no	—
183	C₂×C₄	C₂×C₇×#41	1	6	yes	—
184	C₂×C₄	C₂×C₇×#41	1	6	yes	—
185	C₂×C₄	C₂×C₇×#41	1	6	yes	—
186	C₂×C₄	C₂×C₇×#41	1	6	yes	—
187	C₂×C₄	C₇×2₋¹⁺⁴	1	6	yes	—
188	C₇	#36608	1	1	yes	characteristic, direct product factor, Sylow
189	C₂²	C₂²×C₇×#41	1	3	yes	—
190	C₂²	C₂²×C₇×#41	1	1	yes	characteristic
191	C₂²	C₂×C₇×2₋¹⁺⁴	1	3	yes	—
192	C₄	—	2	6	no	—
193	C₄	—	2	6	no	—
194	C₄	—	4	4	no	—
195	C₄	—	4	12	no	—
196	C₄	—	4	6	no	—
197	C₄	C₇×#556	1	12	yes	—
198	C₂	C₇×#3444	1	1	yes	characteristic
199	C₂	C₂×C₇×#556	1	3	yes	—
200	C₂	C₇×#3509	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₇
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
27. #36608	C₇	direct (N × H)
188. C₇	#36608	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×#2862	2
C₇×#2581	2
C₇×#2514	2
C₇×#2356	2
C₄×C₇×C₄⋊Q₈	2
C₄²×C₇×Q₈	2
#36608	7

Subgroups — C7×#36608 #13213824

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#36608 #13213824