Subgroups — C₇×#2989 #11742298

Order 896.

388 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×#41	C₂	1	1	yes	maximal, characteristic
2	C₇×#446	C₂	1	2	yes	maximal, complemented
3	C₇×#436	C₂	1	2	yes	maximal, complemented
4	C₇×#435	C₂	1	4	yes	maximal, complemented
5	C₇×#432	C₂	1	2	yes	maximal, complemented
6	C₇×#431	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#423	C₂	1	2	yes	maximal, complemented
8	C₂×C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₇×#41	C₄	1	1	yes	characteristic
10	C₂×C₇×M₁₆	C₂²	1	4	yes	—
11	C₂²×C₇×C₈	C₂²	1	2	yes	—
12	C₄×C₇×Q₈	C₄	1	1	yes	characteristic
13	C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
14	C₄×C₇×D₄	C₄	1	2	yes	—
15	C₄×C₇×D₄	C₄	1	1	yes	characteristic
16	C₄×C₇×D₄	C₂²	1	2	yes	—
17	C₄×C₇×D₄	C₂²	1	1	yes	characteristic
18	C₇×#117	C₄	1	2	yes	—
19	C₇×#117	C₄	1	1	yes	characteristic
20	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₇	C₂²	1	2	yes	—
22	C₇×C₄⋊C₈	C₂²	1	2	yes	—
23	C₇×C₄⋊C₈	C₂²	1	2	yes	—
24	C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₇×C₄⋊C₈	C₂²	1	4	yes	—
26	C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₇×C₂²⋊C₈	C₂²	1	2	yes	—
28	C₇×C₂²⋊C₈	C₂²	1	4	yes	—
29	C₇×#97	C₂²	1	4	yes	—
30	C₄×C₇×C₈	C₂²	1	2	yes	—
31	#2989	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₇×#41	—	2	4	no	—
33	C₂×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₇×D₄	C₂×C₄	1	2	yes	—
35	C₂×C₇×D₄	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₇	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₇	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₇	C₂³	1	2	yes	—
42	C₇×M₁₆	—	4	4	no	—
43	C₂×C₇×C₈	—	4	2	no	—
44	C₂×C₇×C₈	C₂³	1	2	yes	—
45	C₂×C₇×C₈	C₂³	1	2	yes	—
46	C₂×C₇×C₈	C₂³	1	4	yes	—
47	C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
48	C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
49	C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
50	C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
51	C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
52	C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
53	C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
54	C₄²×C₇	C₂×C₄	1	1	yes	characteristic
55	C₄²×C₇	C₂×C₄	1	2	yes	—
56	C₄²×C₇	C₂×C₄	1	2	yes	—
57	C₄²×C₇	C₂×C₄	1	1	yes	characteristic
58	C₄²×C₇	C₂³	1	1	yes	characteristic
59	C₄²×C₇	C₂³	1	2	yes	—
60	C₄²×C₇	C₂³	1	1	yes	characteristic
61	C₄×#41	C₂×C₇	1	1	yes	characteristic
62	#446	C₂×C₇	1	2	yes	complemented
63	#436	C₂×C₇	1	2	yes	complemented
64	#435	C₂×C₇	1	4	yes	complemented
65	#432	C₂×C₇	1	2	yes	complemented
66	#431	C₂×C₇	1	1	yes	characteristic, complemented
67	#423	C₂×C₇	1	2	yes	complemented
68	C₂×C₄⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
69	C₂³×C₇	C₂²×C₄	1	2	yes	—
70	C₂³×C₇	C₂²×C₄	1	1	yes	characteristic
71	C₇×Q₈	—	2	2	no	—
72	C₇×D₄	—	2	4	no	—
73	C₇×D₄	—	2	2	no	—
74	C₂×C₄×C₇	—	2	4	no	—
75	C₂×C₄×C₇	—	2	1	no	—
76	C₂×C₄×C₇	—	2	2	no	—
77	C₂×C₄×C₇	—	2	1	no	—
78	C₂×C₄×C₇	—	2	2	no	—
79	C₂×C₄×C₇	—	4	1	no	—
80	C₂×C₄×C₇	—	4	2	no	—
81	C₂×C₄×C₇	M₁₆	1	4	yes	complemented
82	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
91	C₇×C₈	—	2	4	no	—
92	C₇×C₈	—	2	2	no	—
93	C₇×C₈	—	2	2	no	—
94	C₂×#41	C₄×C₇	1	1	yes	characteristic
95	C₂×M₁₆	C₂²×C₇	1	4	yes	—
96	C₂²×C₈	C₂²×C₇	1	2	yes	—
97	C₄×Q₈	C₄×C₇	1	1	yes	characteristic
98	C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
99	C₄×D₄	C₄×C₇	1	2	yes	—
100	C₄×D₄	C₄×C₇	1	1	yes	characteristic
101	C₄×D₄	C₂²×C₇	1	2	yes	—
102	C₄×D₄	C₂²×C₇	1	1	yes	characteristic
103	#117	C₄×C₇	1	2	yes	—
104	#117	C₄×C₇	1	1	yes	characteristic
105	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
106	C₂×C₄²	C₂²×C₇	1	2	yes	—
107	C₄⋊C₈	C₂²×C₇	1	2	yes	—
108	C₄⋊C₈	C₂²×C₇	1	2	yes	—
109	C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
110	C₄⋊C₈	C₂²×C₇	1	4	yes	—
111	C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
112	C₂²⋊C₈	C₂²×C₇	1	2	yes	—
113	C₂²⋊C₈	C₂²×C₇	1	4	yes	—
114	#97	C₂²×C₇	1	4	yes	—
115	C₄×C₈	C₂²×C₇	1	2	yes	—
116	C₂²×C₇	—	2	1	no	—
117	C₂²×C₇	—	2	1	no	—
118	C₂²×C₇	—	2	2	no	—
119	C₂²×C₇	—	2	2	no	—
120	C₂²×C₇	—	2	2	no	—
121	C₂²×C₇	C₂×M₁₆	1	2	yes	—
122	C₂²×C₇	C₂³×C₄	1	1	yes	characteristic
123	C₄×C₇	—	2	1	no	—
124	C₄×C₇	—	2	2	no	—
125	C₄×C₇	—	2	2	no	—
126	C₄×C₇	—	2	1	no	—
127	C₄×C₇	—	2	1	no	—
128	C₄×C₇	—	2	2	no	—
129	C₄×C₇	C₂×M₁₆	1	2	yes	—
130	C₄×C₇	C₂×M₁₆	1	2	yes	—
131	C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
132	C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
133	#41	—	2	4	no	—
134	C₂×Q₈	C₂×C₄×C₇	1	1	yes	characteristic
135	C₂×D₄	C₂×C₄×C₇	1	2	yes	—
136	C₂×D₄	C₂×C₄×C₇	1	1	yes	characteristic
137	C₂²×C₄	C₂×C₄×C₇	1	2	yes	—
138	C₂²×C₄	C₂×C₄×C₇	1	2	yes	—
139	C₂²×C₄	C₂×C₄×C₇	1	1	yes	characteristic
140	C₂²×C₄	C₂×C₄×C₇	1	1	yes	characteristic
141	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
142	C₂²×C₄	C₂³×C₇	1	2	yes	—
143	M₁₆	—	4	4	no	—
144	C₂×C₈	—	4	2	no	—
145	C₂×C₈	C₂³×C₇	1	2	yes	—
146	C₂×C₈	C₂³×C₇	1	2	yes	—
147	C₂×C₈	C₂³×C₇	1	4	yes	—
148	C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
149	C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
150	C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
151	C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
152	C₂²⋊C₄	C₂×C₄×C₇	1	2	yes	—
153	C₂²⋊C₄	C₂×C₄×C₇	1	2	yes	—
154	C₂²⋊C₄	C₂×C₄×C₇	1	2	yes	—
155	C₄²	C₂×C₄×C₇	1	1	yes	characteristic
156	C₄²	C₂×C₄×C₇	1	2	yes	—
157	C₄²	C₂×C₄×C₇	1	2	yes	—
158	C₄²	C₂×C₄×C₇	1	1	yes	characteristic
159	C₄²	C₂³×C₇	1	1	yes	characteristic
160	C₄²	C₂³×C₇	1	2	yes	—
161	C₄²	C₂³×C₇	1	1	yes	characteristic
162	C₂×C₇	—	2	2	no	—
163	C₂×C₇	—	4	2	no	—
164	C₂×C₇	#520	1	1	yes	characteristic
165	C₂×C₇	C₂²×M₁₆	1	1	yes	characteristic
166	C₂×C₇	#568	1	1	yes	characteristic
167	C₂³	C₂²×C₄×C₇	1	2	yes	—
168	C₂³	C₂²×C₄×C₇	1	1	yes	characteristic
169	Q₈	—	2	2	no	—
170	D₄	—	2	4	no	—
171	D₄	—	2	2	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	4	1	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	C₇×M₁₆	1	4	yes	complemented
180	C₂×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
181	C₂×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
182	C₂×C₄	C₂²×C₄×C₇	1	2	yes	—
183	C₂×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
184	C₂×C₄	C₂²×C₄×C₇	1	2	yes	—
185	C₂×C₄	C₂²×C₄×C₇	1	2	yes	—
186	C₂×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
187	C₂×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
188	C₂×C₄	C₂⁴×C₇	1	1	yes	characteristic
189	C₈	—	2	4	no	—
190	C₈	—	2	2	no	—
191	C₈	—	2	2	no	—
192	C₇	#2989	1	1	yes	characteristic, direct product factor, Sylow
193	C₂²	—	2	1	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	2	2	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	2	2	no	—
198	C₂²	C₂×C₇×M₁₆	1	2	yes	—
199	C₂²	C₂³×C₄×C₇	1	1	yes	characteristic
200	C₄	—	2	1	no	—
201	C₄	—	2	2	no	—
202	C₄	—	2	2	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	2	2	no	—
206	C₄	C₂×C₇×M₁₆	1	2	yes	—
207	C₄	C₂×C₇×M₁₆	1	2	yes	—
208	C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
209	C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
210	C₂	—	2	2	no	—
211	C₂	—	4	2	no	—
212	C₂	C₇×#520	1	1	yes	characteristic
213	C₂	C₂²×C₇×M₁₆	1	1	yes	characteristic
214	C₂	C₇×#568	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂⁴
commutator	C₂²	C₂³×C₄×C₇
Frattini, cosocle	C₂×C₄	C₂⁴×C₇
socle	C₂²×C₇	C₂³×C₄
prime radical	C₂×C₇×#41	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#446	C₂	semidirect
3. C₇×#436	C₂	semidirect
4. C₇×#435	C₂	semidirect
5. C₇×#432	C₂	semidirect
6. C₇×#431	C₂	semidirect
7. C₇×#423	C₂	semidirect
8. C₂×C₇×C₄⋊C₈	C₂	semidirect
31. #2989	C₇	direct (N × H)
62. #446	C₂×C₇	semidirect
63. #436	C₂×C₇	semidirect
64. #435	C₂×C₇	semidirect
65. #432	C₂×C₇	semidirect
66. #431	C₂×C₇	semidirect
67. #423	C₂×C₇	semidirect
68. C₂×C₄⋊C₈	C₂×C₇	semidirect
81. C₂×C₄×C₇	M₁₆	semidirect
179. C₂×C₄	C₇×M₁₆	semidirect
192. C₇	#2989	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×C₇×#41	2
C₇×#446	2
C₇×#436	2
C₇×#435	2
C₇×#432	2
C₇×#431	2
C₇×#423	2
C₂×C₇×C₄⋊C₈	2
#2989	7

Subgroups — C7×#2989 #11742298

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2989 #11742298