Subgroups — C₇×#3299 #11742608

Order 896.

388 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#554	C₂	1	1	yes	maximal, characteristic
2	C₇×D₄×Q₈	C₂	1	1	yes	maximal, characteristic
3	C₇×#487	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#484	C₂	1	2	yes	maximal, complemented
5	C₇×#477	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#467	C₂	1	2	yes	maximal, complemented
7	C₇×#462	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#451	C₂	1	2	yes	maximal, complemented
9	C₄×C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×C₈×D₄	C₂	1	1	yes	maximal, characteristic
11	C₂×C₇×#103	C₂	1	2	yes	maximal, complemented
12	C₂²×C₇×Q₈	C₂²	1	2	yes	—
13	C₂×C₇×Q₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₇×Q₁₆	C₂²	1	2	yes	—
15	C₂²×C₇×C₈	C₂²	1	2	yes	—
16	C₇×C₄⋊Q₈	—	2	1	no	—
17	C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₇×#125	—	2	1	no	—
20	C₇×C₂²⋊Q₈	—	2	2	no	—
21	C₇×C₂²⋊Q₈	—	2	2	no	—
22	C₇×C₂²⋊Q₈	C₂²	1	2	yes	—
23	C₇×C₂²⋊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×D₄	—	2	1	no	—
27	C₄×C₇×D₄	—	2	1	no	—
28	C₄×C₇×D₄	C₂²	1	1	yes	characteristic
29	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
30	C₇×#107	C₂²	1	2	yes	—
31	C₇×#107	C₂²	1	1	yes	characteristic
32	C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
33	C₇×#103	C₂²	1	1	yes	characteristic
34	C₇×#103	C₂²	1	2	yes	—
35	C₇×#103	C₂²	1	2	yes	—
36	C₇×#103	C₂²	1	2	yes	—
37	C₇×#103	C₂²	1	2	yes	—
38	C₇×#103	C₂²	1	1	yes	characteristic
39	C₇×C₂²⋊C₈	C₂²	1	2	yes	—
40	C₄×C₇×C₈	C₂²	1	1	yes	characteristic
41	#3299	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂×C₇×Q₈	—	2	2	no	—
43	C₂×C₇×Q₈	—	2	1	no	—
44	C₂×C₇×Q₈	—	2	4	no	—
45	C₂×C₇×Q₈	—	4	2	no	—
46	C₂×C₇×Q₈	C₂³	1	2	yes	—
47	C₂×C₇×Q₈	C₂³	1	1	yes	characteristic
48	C₂×C₇×Q₈	C₂³	1	1	yes	characteristic
49	C₂×C₇×D₄	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₇	—	2	2	no	—
51	C₂²×C₄×C₇	—	2	2	no	—
52	C₂²×C₄×C₇	C₂³	1	2	yes	—
53	C₇×Q₁₆	—	2	2	no	—
54	C₇×Q₁₆	—	4	2	no	—
55	C₂×C₇×C₈	—	4	2	no	—
56	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
57	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
58	C₂×C₇×C₈	C₂³	1	2	yes	—
59	C₇×C₄⋊C₄	—	2	2	no	—
60	C₇×C₄⋊C₄	—	2	1	no	—
61	C₇×C₄⋊C₄	—	2	1	no	—
62	C₇×C₄⋊C₄	—	2	1	no	—
63	C₇×C₄⋊C₄	—	2	1	no	—
64	C₇×C₄⋊C₄	—	2	1	no	—
65	C₇×C₄⋊C₄	—	2	2	no	—
66	C₇×C₄⋊C₄	—	2	2	no	—
67	C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
68	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
69	C₇×C₄⋊C₄	C₂³	1	2	yes	—
70	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
71	C₇×C₂²⋊C₄	—	2	2	no	—
72	C₇×C₂²⋊C₄	—	2	2	no	—
73	C₇×C₂²⋊C₄	D₄	1	2	yes	—
74	C₄²×C₇	—	2	1	no	—
75	C₄²×C₇	—	2	1	no	—
76	C₄²×C₇	C₂³	1	1	yes	characteristic
77	#554	C₂×C₇	1	1	yes	characteristic
78	D₄×Q₈	C₂×C₇	1	1	yes	characteristic
79	#487	C₂×C₇	1	1	yes	characteristic, complemented
80	#484	C₂×C₇	1	2	yes	complemented
81	#477	C₂×C₇	1	1	yes	characteristic, complemented
82	#467	C₂×C₇	1	2	yes	complemented
83	#462	C₂×C₇	1	1	yes	characteristic, complemented
84	#451	C₂×C₇	1	2	yes	complemented
85	C₄×Q₁₆	C₂×C₇	1	1	yes	characteristic, complemented
86	C₈×D₄	C₂×C₇	1	1	yes	characteristic
87	C₂×#103	C₂×C₇	1	2	yes	complemented
88	C₂³×C₇	C₂×D₄	1	2	yes	—
89	C₇×Q₈	—	2	1	no	—
90	C₇×Q₈	—	2	1	no	—
91	C₇×Q₈	—	2	2	no	—
92	C₇×Q₈	—	2	1	no	—
93	C₇×Q₈	—	2	2	no	—
94	C₇×Q₈	—	4	4	no	—
95	C₇×Q₈	—	8	1	no	—
96	C₇×Q₈	#41	1	2	yes	—
97	C₇×D₄	Q₁₆	1	4	yes	complemented
98	C₂×C₄×C₇	—	2	1	no	—
99	C₂×C₄×C₇	—	2	2	no	—
100	C₂×C₄×C₇	—	2	2	no	—
101	C₂×C₄×C₇	—	2	1	no	—
102	C₂×C₄×C₇	—	2	1	no	—
103	C₂×C₄×C₇	—	2	1	no	—
104	C₂×C₄×C₇	—	2	2	no	—
105	C₂×C₄×C₇	—	2	2	no	—
106	C₂×C₄×C₇	—	4	4	no	—
107	C₂×C₄×C₇	—	8	2	no	—
108	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
112	C₇×C₈	—	2	1	no	—
113	C₇×C₈	—	2	1	no	—
114	C₇×C₈	—	2	2	no	—
115	C₂²×Q₈	C₂²×C₇	1	2	yes	—
116	C₂×Q₁₆	C₂²×C₇	1	1	yes	characteristic
117	C₂×Q₁₆	C₂²×C₇	1	2	yes	—
118	C₂²×C₈	C₂²×C₇	1	2	yes	—
119	C₄⋊Q₈	—	2	1	no	—
120	C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
121	C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
122	#125	—	2	1	no	—
123	C₂²⋊Q₈	—	2	2	no	—
124	C₂²⋊Q₈	—	2	2	no	—
125	C₂²⋊Q₈	C₂²×C₇	1	2	yes	—
126	C₂²⋊Q₈	C₂²×C₇	1	2	yes	—
127	C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
128	C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
129	C₄×D₄	—	2	1	no	—
130	C₄×D₄	—	2	1	no	—
131	C₄×D₄	C₂²×C₇	1	1	yes	characteristic
132	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
133	#107	C₂²×C₇	1	1	yes	characteristic
134	#107	C₂²×C₇	1	2	yes	—
135	C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
136	#103	C₂²×C₇	1	1	yes	characteristic
137	#103	C₂²×C₇	1	2	yes	—
138	#103	C₂²×C₇	1	2	yes	—
139	#103	C₂²×C₇	1	2	yes	—
140	#103	C₂²×C₇	1	1	yes	characteristic
141	#103	C₂²×C₇	1	2	yes	—
142	C₂²⋊C₈	C₂²×C₇	1	2	yes	—
143	C₄×C₈	C₂²×C₇	1	1	yes	characteristic
144	C₂²×C₇	—	2	2	no	—
145	C₂²×C₇	—	2	2	no	—
146	C₂²×C₇	C₂×Q₁₆	1	4	yes	—
147	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
148	C₄×C₇	—	2	2	no	—
149	C₄×C₇	—	2	2	no	—
150	C₄×C₇	—	2	1	no	—
151	C₄×C₇	—	4	2	no	—
152	C₄×C₇	—	4	1	no	—
153	C₄×C₇	—	4	1	no	—
154	C₄×C₇	—	4	1	no	—
155	C₄×C₇	—	4	2	no	—
156	C₄×C₇	C₂×Q₁₆	1	2	yes	—
157	C₄×C₇	C₂×#41	1	1	yes	characteristic
158	C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
159	C₂×Q₈	—	2	2	no	—
160	C₂×Q₈	—	2	1	no	—
161	C₂×Q₈	—	2	4	no	—
162	C₂×Q₈	—	4	2	no	—
163	C₂×Q₈	C₂³×C₇	1	2	yes	—
164	C₂×Q₈	C₂³×C₇	1	1	yes	characteristic
165	C₂×Q₈	C₂³×C₇	1	1	yes	characteristic
166	C₂×D₄	C₇×D₄	1	1	yes	characteristic
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	C₂³×C₇	1	2	yes	—
170	Q₁₆	—	2	2	no	—
171	Q₁₆	—	4	2	no	—
172	C₂×C₈	—	4	2	no	—
173	C₂×C₈	C₂³×C₇	1	1	yes	characteristic
174	C₂×C₈	C₂³×C₇	1	1	yes	characteristic
175	C₂×C₈	C₂³×C₇	1	2	yes	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	1	no	—
178	C₄⋊C₄	—	2	1	no	—
179	C₄⋊C₄	—	2	1	no	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
185	C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
186	C₄⋊C₄	C₂³×C₇	1	2	yes	—
187	C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
188	C₂²⋊C₄	—	2	2	no	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	C₇×D₄	1	2	yes	—
191	C₄²	—	2	1	no	—
192	C₄²	—	2	1	no	—
193	C₄²	C₂³×C₇	1	1	yes	characteristic
194	C₂×C₇	—	2	4	no	—
195	C₂×C₇	#546	1	1	yes	characteristic
196	C₂×C₇	C₂²×Q₁₆	1	1	yes	characteristic
197	C₂×C₇	#577	1	1	yes	characteristic
198	C₂³	C₂×C₇×D₄	1	2	yes	—
199	Q₈	—	2	1	no	—
200	Q₈	—	2	1	no	—
201	Q₈	—	2	2	no	—
202	Q₈	—	2	1	no	—
203	Q₈	—	2	2	no	—
204	Q₈	—	4	4	no	—
205	Q₈	—	8	1	no	—
206	Q₈	C₇×#41	1	2	yes	—
207	D₄	C₇×Q₁₆	1	4	yes	complemented
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	8	2	no	—
218	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
219	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
220	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
221	C₂×C₄	C₂⁴×C₇	1	1	yes	characteristic
222	C₈	—	2	1	no	—
223	C₈	—	2	1	no	—
224	C₈	—	2	2	no	—
225	C₇	#3299	1	1	yes	characteristic, direct product factor, Sylow
226	C₂²	—	2	2	no	—
227	C₂²	—	2	2	no	—
228	C₂²	C₂×C₇×Q₁₆	1	4	yes	—
229	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
230	C₄	—	2	2	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	1	no	—
233	C₄	—	4	2	no	—
234	C₄	—	4	1	no	—
235	C₄	—	4	1	no	—
236	C₄	—	4	2	no	—
237	C₄	—	4	1	no	—
238	C₄	C₂×C₇×Q₁₆	1	2	yes	—
239	C₄	C₂×C₇×#41	1	1	yes	characteristic
240	C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
241	C₂	—	2	4	no	—
242	C₂	C₇×#546	1	1	yes	characteristic
243	C₂	C₂²×C₇×Q₁₆	1	1	yes	characteristic
244	C₂	C₇×#577	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₇×#487	C₂	semidirect
4. C₇×#484	C₂	semidirect
5. C₇×#477	C₂	semidirect
6. C₇×#467	C₂	semidirect
7. C₇×#462	C₂	semidirect
8. C₇×#451	C₂	semidirect
9. C₄×C₇×Q₁₆	C₂	semidirect
11. C₂×C₇×#103	C₂	semidirect
41. #3299	C₇	direct (N × H)
79. #487	C₂×C₇	semidirect
80. #484	C₂×C₇	semidirect
81. #477	C₂×C₇	semidirect
82. #467	C₂×C₇	semidirect
83. #462	C₂×C₇	semidirect
84. #451	C₂×C₇	semidirect
85. C₄×Q₁₆	C₂×C₇	semidirect
87. C₂×#103	C₂×C₇	semidirect
97. C₇×D₄	Q₁₆	semidirect
207. D₄	C₇×Q₁₆	semidirect
225. C₇	#3299	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×#554	2
C₇×D₄×Q₈	2
C₇×#487	2
C₇×#484	2
C₇×#477	2
C₇×#467	2
C₇×#462	2
C₇×#451	2
C₄×C₇×Q₁₆	2
C₇×C₈×D₄	2
C₂×C₇×#103	2
#3299	7

Subgroups — C7×#3299 #11742608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#3299 #11742608