Subgroups — C₇×#2799 #11742108

Order 896.

376 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#400	C₂	1	2	yes	maximal
2	C₇×#400	C₂	1	1	yes	maximal, characteristic
3	C₇×#398	C₂	1	2	yes	maximal
4	C₇×#395	C₂	1	1	yes	maximal, characteristic
5	C₇×#391	C₂	1	2	yes	maximal
6	C₇×#391	C₂	1	2	yes	maximal
7	C₇×#391	C₂	1	1	yes	maximal, characteristic
8	C₇×#387	C₂	1	2	yes	maximal
9	C₇×#387	C₂	1	2	yes	maximal
10	C₂²×C₇×Q₈	C₂²	1	1	yes	characteristic
11	C₂²×C₇×Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₇	C₂²	1	2	yes	—
20	C₇×#95	C₂²	1	2	yes	—
21	C₇×#95	C₂²	1	2	yes	—
22	C₇×#95	C₂²	1	2	yes	—
23	C₇×#95	C₂²	1	2	yes	—
24	C₇×#95	C₂²	1	2	yes	—
25	C₇×#95	C₂²	1	2	yes	—
26	C₇×#95	C₂²	1	2	yes	—
27	C₇×#95	C₂²	1	2	yes	—
28	C₇×#95	C₂²	1	2	yes	—
29	C₇×#95	C₂²	1	1	yes	characteristic
30	C₇×#95	C₂²	1	1	yes	characteristic
31	#2799	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂×C₇×Q₈	—	2	4	no	—
33	C₂×C₇×Q₈	—	4	1	no	—
34	C₂×C₇×Q₈	—	4	1	no	—
35	C₂×C₇×Q₈	—	4	1	no	—
36	C₂×C₇×Q₈	Q₈	1	4	yes	complemented
37	C₂²×C₄×C₇	C₂³	1	2	yes	—
38	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₇	C₂³	1	2	yes	—
40	C₂²×C₄×C₇	C₂³	1	2	yes	—
41	C₂²×C₄×C₇	C₂³	1	2	yes	—
42	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₇	C₂³	1	2	yes	—
45	C₂²×C₄×C₇	C₂³	1	2	yes	—
46	C₇×C₄⋊C₄	—	2	4	no	—
47	C₇×C₄⋊C₄	—	2	4	no	—
48	C₇×C₄⋊C₄	—	4	2	no	—
49	C₇×C₄⋊C₄	—	4	1	no	—
50	C₇×C₄⋊C₄	—	4	1	no	—
51	C₇×C₄⋊C₄	—	4	2	no	—
52	C₄²×C₇	—	4	2	no	—
53	C₄²×C₇	—	4	1	no	—
54	#400	C₂×C₇	1	2	yes	—
55	#400	C₂×C₇	1	1	yes	characteristic
56	#398	C₂×C₇	1	2	yes	—
57	#395	C₂×C₇	1	1	yes	characteristic
58	#391	C₂×C₇	1	2	yes	—
59	#391	C₂×C₇	1	2	yes	—
60	#391	C₂×C₇	1	1	yes	characteristic
61	#387	C₂×C₇	1	2	yes	—
62	#387	C₂×C₇	1	2	yes	—
63	C₂³×C₇	C₂⁴	1	1	yes	characteristic
64	C₇×Q₈	—	4	4	no	—
65	C₇×Q₈	—	4	4	no	—
66	C₂×C₄×C₇	—	2	2	no	—
67	C₂×C₄×C₇	—	2	1	no	—
68	C₂×C₄×C₇	—	2	2	no	—
69	C₂×C₄×C₇	—	2	2	no	—
70	C₂×C₄×C₇	—	2	2	no	—
71	C₂×C₄×C₇	—	2	1	no	—
72	C₂×C₄×C₇	—	2	2	no	—
73	C₂×C₄×C₇	—	2	2	no	—
74	C₂×C₄×C₇	—	2	2	no	—
75	C₂×C₄×C₇	—	2	2	no	—
76	C₂×C₄×C₇	—	2	2	no	—
77	C₂×C₄×C₇	—	2	2	no	—
78	C₂×C₄×C₇	—	2	2	no	—
79	C₂×C₄×C₇	—	2	2	no	—
80	C₂×C₄×C₇	—	2	2	no	—
81	C₂×C₄×C₇	—	2	1	no	—
82	C₂×C₄×C₇	—	2	1	no	—
83	C₂×C₄×C₇	—	2	2	no	—
84	C₂×C₄×C₇	—	2	2	no	—
85	C₂×C₄×C₇	—	2	2	no	—
86	C₂×C₄×C₇	—	2	4	no	—
87	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
88	C₂×C₄×C₇	C₂×Q₈	1	4	yes	—
89	C₂×C₄×C₇	#41	1	4	yes	—
90	C₂²×Q₈	C₂²×C₇	1	1	yes	characteristic
91	C₂²×Q₈	C₂²×C₇	1	1	yes	characteristic
92	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
93	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
94	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
95	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
96	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
97	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
98	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
99	C₂×C₄²	C₂²×C₇	1	2	yes	—
100	#95	C₂²×C₇	1	2	yes	—
101	#95	C₂²×C₇	1	2	yes	—
102	#95	C₂²×C₇	1	2	yes	—
103	#95	C₂²×C₇	1	2	yes	—
104	#95	C₂²×C₇	1	2	yes	—
105	#95	C₂²×C₇	1	2	yes	—
106	#95	C₂²×C₇	1	2	yes	—
107	#95	C₂²×C₇	1	2	yes	—
108	#95	C₂²×C₇	1	2	yes	—
109	#95	C₂²×C₇	1	1	yes	characteristic
110	#95	C₂²×C₇	1	1	yes	characteristic
111	C₂²×C₇	C₂²×Q₈	1	1	yes	characteristic
112	C₂²×C₇	C₂×#41	1	2	yes	—
113	C₂²×C₇	H(5,𝔽₂)	1	1	yes	characteristic
114	C₂²×C₇	H(5,𝔽₂)	1	1	yes	characteristic
115	C₂²×C₇	2₋¹⁺⁴	1	2	yes	—
116	C₄×C₇	—	2	2	no	—
117	C₄×C₇	—	2	4	no	—
118	C₄×C₇	—	2	4	no	—
119	C₄×C₇	—	4	2	no	—
120	C₄×C₇	—	4	2	no	—
121	C₄×C₇	—	4	2	no	—
122	C₄×C₇	—	4	1	no	—
123	C₄×C₇	—	4	1	no	—
124	C₄×C₇	—	4	2	no	—
125	C₂×Q₈	—	2	4	no	—
126	C₂×Q₈	—	4	1	no	—
127	C₂×Q₈	—	4	1	no	—
128	C₂×Q₈	—	4	1	no	—
129	C₂×Q₈	C₇×Q₈	1	4	yes	complemented
130	C₂²×C₄	C₂³×C₇	1	2	yes	—
131	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
132	C₂²×C₄	C₂³×C₇	1	2	yes	—
133	C₂²×C₄	C₂³×C₇	1	2	yes	—
134	C₂²×C₄	C₂³×C₇	1	2	yes	—
135	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
136	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
137	C₂²×C₄	C₂³×C₇	1	2	yes	—
138	C₂²×C₄	C₂³×C₇	1	2	yes	—
139	C₄⋊C₄	—	2	4	no	—
140	C₄⋊C₄	—	2	4	no	—
141	C₄⋊C₄	—	4	2	no	—
142	C₄⋊C₄	—	4	1	no	—
143	C₄⋊C₄	—	4	1	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄²	—	4	2	no	—
146	C₄²	—	4	1	no	—
147	C₂×C₇	#542	1	2	yes	—
148	C₂×C₇	#543	1	1	yes	characteristic
149	C₂×C₇	#551	1	1	yes	characteristic
150	C₂×C₇	#557	1	2	yes	—
151	C₂×C₇	#563	1	1	yes	characteristic
152	C₂³	C₂⁴×C₇	1	1	yes	characteristic
153	Q₈	—	4	4	no	—
154	Q₈	—	4	4	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	C₂×C₇×Q₈	1	2	yes	—
177	C₂×C₄	C₂×C₇×Q₈	1	4	yes	—
178	C₂×C₄	C₇×#41	1	4	yes	—
179	C₇	#2799	1	1	yes	characteristic, direct product factor, Sylow
180	C₂²	C₂²×C₇×Q₈	1	1	yes	characteristic
181	C₂²	C₂×C₇×#41	1	2	yes	—
182	C₂²	C₇×H(5,𝔽₂)	1	1	yes	characteristic
183	C₂²	C₇×H(5,𝔽₂)	1	1	yes	characteristic
184	C₂²	C₇×2₋¹⁺⁴	1	2	yes	—
185	C₄	—	2	2	no	—
186	C₄	—	2	4	no	—
187	C₄	—	2	4	no	—
188	C₄	—	4	2	no	—
189	C₄	—	4	2	no	—
190	C₄	—	4	2	no	—
191	C₄	—	4	1	no	—
192	C₄	—	4	1	no	—
193	C₄	—	4	2	no	—
194	C₂	C₇×#542	1	2	yes	—
195	C₂	C₇×#543	1	1	yes	characteristic
196	C₂	C₇×#551	1	1	yes	characteristic
197	C₂	C₇×#557	1	2	yes	—
198	C₂	C₇×#563	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×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
31. #2799	C₇	direct (N × H)
36. C₂×C₇×Q₈	Q₈	semidirect
129. C₂×Q₈	C₇×Q₈	semidirect
179. C₇	#2799	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×#400	2
C₇×#398	2
C₇×#395	2
C₇×#391	2
C₇×#387	2
#2799	7

Subgroups — C7×#2799 #11742108

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2799 #11742108