Subgroups — C₇×#3387 #11742696

Order 896.

326 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×Q₈²	C₂	1	1	yes	maximal, characteristic
2	C₇×#557	C₂	1	1	yes	maximal, characteristic
3	C₇×#501	C₂	1	2	yes	maximal
4	C₇×#498	C₂	1	1	yes	maximal, characteristic
5	C₇×#479	C₂	1	2	yes	maximal
6	C₇×#477	C₂	1	1	yes	maximal, characteristic
7	C₇×#475	C₂	1	2	yes	maximal
8	C₇×#475	C₂	1	1	yes	maximal, characteristic
9	C₇×#446	C₂	1	1	yes	maximal, characteristic
10	C₇×#441	C₂	1	2	yes	maximal
11	C₄×C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic
12	C₂×C₇×Q₁₆	C₂²	1	1	yes	characteristic
13	C₇×C₄⋊Q₈	—	2	1	no	—
14	C₇×C₄⋊Q₈	—	2	1	no	—
15	C₇×C₄⋊Q₈	—	2	2	no	—
16	C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₇×C₄⋊Q₈	C₂²	1	2	yes	—
18	C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₇×#125	—	2	2	no	—
20	C₇×#125	C₂²	1	2	yes	—
21	C₄×C₇×Q₈	—	2	1	no	—
22	C₄×C₇×Q₈	—	2	1	no	—
23	C₄×C₇×Q₈	C₂²	1	2	yes	—
24	C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
25	C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
26	C₄×C₇×Q₈	C₂²	1	2	yes	—
27	C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
28	C₇×#107	C₂²	1	2	yes	—
29	C₇×#107	C₂²	1	1	yes	characteristic
30	C₇×#106	C₂²	1	2	yes	—
31	C₇×#106	C₂²	1	2	yes	—
32	C₇×#106	C₂²	1	2	yes	—
33	C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
34	C₇×C₄⋊C₈	C₂²	1	2	yes	—
35	C₇×#103	C₂²	1	1	yes	characteristic
36	C₇×#103	C₂²	1	2	yes	—
37	C₇×#103	C₂²	1	2	yes	—
38	C₇×#103	C₂²	1	1	yes	characteristic
39	C₇×#97	C₂²	1	2	yes	—
40	C₄×C₇×C₈	C₂²	1	1	yes	characteristic
41	#3387	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂×C₇×Q₈	—	2	1	no	—
43	C₂×C₇×Q₈	—	2	2	no	—
44	C₂×C₇×Q₈	—	2	1	no	—
45	C₂×C₇×Q₈	D₄	1	1	yes	characteristic
46	C₂×C₇×Q₈	C₂³	1	1	yes	characteristic
47	C₂×C₇×Q₈	C₂³	1	1	yes	characteristic
48	C₇×Q₁₆	Q₈	1	4	yes	complemented
49	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
50	C₂×C₇×C₈	C₂³	1	2	yes	—
51	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
52	C₇×C₄⋊C₄	—	2	2	no	—
53	C₇×C₄⋊C₄	—	2	2	no	—
54	C₇×C₄⋊C₄	—	2	1	no	—
55	C₇×C₄⋊C₄	—	2	1	no	—
56	C₇×C₄⋊C₄	—	2	2	no	—
57	C₇×C₄⋊C₄	—	2	2	no	—
58	C₇×C₄⋊C₄	—	2	1	no	—
59	C₇×C₄⋊C₄	—	2	1	no	—
60	C₇×C₄⋊C₄	—	2	2	no	—
61	C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
62	C₇×C₄⋊C₄	D₄	1	2	yes	—
63	C₇×C₄⋊C₄	C₂³	1	2	yes	—
64	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
65	C₇×C₄⋊C₄	C₂³	1	2	yes	—
66	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
67	C₄²×C₇	—	2	2	no	—
68	C₄²×C₇	—	2	1	no	—
69	C₄²×C₇	—	2	2	no	—
70	C₄²×C₇	—	2	1	no	—
71	C₄²×C₇	C₂³	1	1	yes	characteristic
72	C₄²×C₇	C₂³	1	2	yes	—
73	Q₈²	C₂×C₇	1	1	yes	characteristic
74	#557	C₂×C₇	1	1	yes	characteristic
75	#501	C₂×C₇	1	2	yes	—
76	#498	C₂×C₇	1	1	yes	characteristic
77	#479	C₂×C₇	1	2	yes	—
78	#477	C₂×C₇	1	1	yes	characteristic
79	#475	C₂×C₇	1	2	yes	—
80	#475	C₂×C₇	1	1	yes	characteristic
81	#446	C₂×C₇	1	1	yes	characteristic
82	#441	C₂×C₇	1	2	yes	—
83	C₄×Q₁₆	C₂×C₇	1	1	yes	characteristic
84	C₇×Q₈	—	2	2	no	—
85	C₇×Q₈	—	2	1	no	—
86	C₇×Q₈	—	2	1	no	—
87	C₇×Q₈	—	4	2	no	—
88	C₇×Q₈	—	8	2	no	—
89	C₇×Q₈	—	8	1	no	—
90	C₇×Q₈	C₂×Q₈	1	2	yes	—
91	C₇×Q₈	C₂×Q₈	1	2	yes	—
92	C₂×C₄×C₇	—	2	1	no	—
93	C₂×C₄×C₇	—	2	1	no	—
94	C₂×C₄×C₇	—	2	1	no	—
95	C₂×C₄×C₇	—	2	2	no	—
96	C₂×C₄×C₇	—	2	2	no	—
97	C₂×C₄×C₇	—	2	1	no	—
98	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
99	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
100	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
103	C₇×C₈	—	2	1	no	—
104	C₇×C₈	—	2	2	no	—
105	C₇×C₈	C₂×Q₈	1	2	yes	—
106	C₂×Q₁₆	C₂²×C₇	1	1	yes	characteristic
107	C₄⋊Q₈	—	2	1	no	—
108	C₄⋊Q₈	—	2	1	no	—
109	C₄⋊Q₈	—	2	2	no	—
110	C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
111	C₄⋊Q₈	C₂²×C₇	1	2	yes	—
112	C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
113	#125	—	2	2	no	—
114	#125	C₂²×C₇	1	2	yes	—
115	C₄×Q₈	—	2	1	no	—
116	C₄×Q₈	—	2	1	no	—
117	C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
118	C₄×Q₈	C₂²×C₇	1	2	yes	—
119	C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
120	C₄×Q₈	C₂²×C₇	1	2	yes	—
121	C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
122	#107	C₂²×C₇	1	1	yes	characteristic
123	#107	C₂²×C₇	1	2	yes	—
124	#106	C₂²×C₇	1	2	yes	—
125	#106	C₂²×C₇	1	2	yes	—
126	#106	C₂²×C₇	1	2	yes	—
127	C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
128	C₄⋊C₈	C₂²×C₇	1	2	yes	—
129	#103	C₂²×C₇	1	1	yes	characteristic
130	#103	C₂²×C₇	1	1	yes	characteristic
131	#103	C₂²×C₇	1	2	yes	—
132	#103	C₂²×C₇	1	2	yes	—
133	#97	C₂²×C₇	1	2	yes	—
134	C₄×C₈	C₂²×C₇	1	1	yes	characteristic
135	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
136	C₄×C₇	—	2	2	no	—
137	C₄×C₇	—	2	2	no	—
138	C₄×C₇	—	2	2	no	—
139	C₄×C₇	—	2	2	no	—
140	C₄×C₇	—	2	1	no	—
141	C₄×C₇	—	4	2	no	—
142	C₄×C₇	—	4	1	no	—
143	C₄×C₇	—	4	2	no	—
144	C₄×C₇	—	4	1	no	—
145	C₄×C₇	#137	1	2	yes	—
146	C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
147	C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
148	C₂×Q₈	—	2	1	no	—
149	C₂×Q₈	—	2	2	no	—
150	C₂×Q₈	—	2	1	no	—
151	C₂×Q₈	C₇×D₄	1	1	yes	characteristic
152	C₂×Q₈	C₂³×C₇	1	1	yes	characteristic
153	C₂×Q₈	C₂³×C₇	1	1	yes	characteristic
154	Q₁₆	C₇×Q₈	1	4	yes	complemented
155	C₂×C₈	C₂³×C₇	1	1	yes	characteristic
156	C₂×C₈	C₂³×C₇	1	2	yes	—
157	C₂×C₈	C₂³×C₇	1	1	yes	characteristic
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	2	1	no	—
161	C₄⋊C₄	—	2	1	no	—
162	C₄⋊C₄	—	2	2	no	—
163	C₄⋊C₄	—	2	2	no	—
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	—	2	1	no	—
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
168	C₄⋊C₄	C₇×D₄	1	2	yes	—
169	C₄⋊C₄	C₂³×C₇	1	2	yes	—
170	C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
171	C₄⋊C₄	C₂³×C₇	1	2	yes	—
172	C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
173	C₄²	—	2	2	no	—
174	C₄²	—	2	1	no	—
175	C₄²	—	2	2	no	—
176	C₄²	—	2	1	no	—
177	C₄²	C₂³×C₇	1	1	yes	characteristic
178	C₄²	C₂³×C₇	1	2	yes	—
179	C₂×C₇	D₄×Q₈	1	1	yes	characteristic
180	C₂×C₇	C₂×#137	1	1	yes	characteristic
181	C₂×C₇	#577	1	1	yes	characteristic
182	Q₈	—	2	2	no	—
183	Q₈	—	2	1	no	—
184	Q₈	—	2	1	no	—
185	Q₈	—	4	2	no	—
186	Q₈	—	8	2	no	—
187	Q₈	—	8	1	no	—
188	Q₈	C₂×C₇×Q₈	1	2	yes	—
189	Q₈	C₂×C₇×Q₈	1	2	yes	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
197	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
198	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
199	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
200	C₂×C₄	C₂⁴×C₇	1	1	yes	characteristic
201	C₈	—	2	1	no	—
202	C₈	—	2	2	no	—
203	C₈	C₂×C₇×Q₈	1	2	yes	—
204	C₇	#3387	1	1	yes	characteristic, direct product factor, Sylow
205	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
206	C₄	—	2	2	no	—
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	2	2	no	—
210	C₄	—	2	1	no	—
211	C₄	—	4	2	no	—
212	C₄	—	4	2	no	—
213	C₄	—	4	1	no	—
214	C₄	—	4	1	no	—
215	C₄	C₇×#137	1	2	yes	—
216	C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
217	C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
218	C₂	C₇×D₄×Q₈	1	1	yes	characteristic
219	C₂	C₂×C₇×#137	1	1	yes	characteristic
220	C₂	C₇×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄×C₇×Q₈ ◁ G

Complements

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

N	H ≅ G/N	type
41. #3387	C₇	direct (N × H)
48. C₇×Q₁₆	Q₈	semidirect
154. Q₁₆	C₇×Q₈	semidirect
204. C₇	#3387	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×Q₈²	2
C₇×#557	2
C₇×#501	2
C₇×#498	2
C₇×#479	2
C₇×#477	2
C₇×#475	2
C₇×#446	2
C₇×#441	2
C₄×C₇×Q₁₆	2
#3387	7

Subgroups — C7×#3387 #11742696

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#3387 #11742696