Subgroups — C₇×#3329 #11742638

Order 896.

386 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#555	C₂	1	1	yes	maximal, characteristic
2	C₇×D₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#487	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#485	C₂	1	2	yes	maximal, complemented
5	C₇×#477	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#468	C₂	1	2	yes	maximal, complemented
7	C₇×#461	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#448	C₂	1	2	yes	maximal, complemented
9	C₄×C₇×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#436	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#419	C₂	1	2	yes	maximal, complemented
12	C₂²×C₇×Q₈	C₂²	1	2	yes	—
13	C₂×C₇×SD₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₇×SD₁₆	C₂²	1	2	yes	—
15	C₂×C₇×M₁₆	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×#124	—	2	2	no	—
20	C₇×C₂²⋊Q₈	—	2	2	no	—
21	C₇×C₂²⋊Q₈	C₂²	1	2	yes	—
22	C₇×#121	—	2	1	no	—
23	C₇×#121	C₂²	1	2	yes	—
24	C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
25	C₄×C₇×D₄	—	2	1	no	—
26	C₄×C₇×D₄	C₂²	1	1	yes	characteristic
27	C₄×C₇×D₄	C₂²	1	1	yes	characteristic
28	C₇×#117	—	2	1	no	—
29	C₇×#117	C₂²	1	2	yes	—
30	C₇×#107	C₂²	1	2	yes	—
31	C₇×#106	C₂²	1	1	yes	characteristic
32	C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
33	C₇×#103	C₂²	1	2	yes	—
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	1	yes	characteristic
38	C₇×#102	C₂²	1	1	yes	characteristic
39	C₇×C₂²⋊C₈	C₂²	1	2	yes	—
40	C₄×C₇×C₈	C₂²	1	1	yes	characteristic
41	#3329	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×D₄	—	2	2	no	—
49	C₂×C₇×D₄	D₄	1	1	yes	characteristic
50	C₂×C₇×D₄	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₇	—	2	2	no	—
52	C₂²×C₄×C₇	—	2	1	no	—
53	C₂²×C₄×C₇	C₂³	1	2	yes	—
54	C₇×SD₁₆	—	2	2	no	—
55	C₇×SD₁₆	—	4	2	no	—
56	C₇×M₁₆	—	4	2	no	—
57	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
58	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
59	C₂×C₇×C₈	C₂³	1	2	yes	—
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	2	no	—
65	C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
66	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
67	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
68	C₇×C₄⋊C₄	C₂³	1	2	yes	—
69	C₇×C₂²⋊C₄	—	2	2	no	—
70	C₇×C₂²⋊C₄	—	2	2	no	—
71	C₇×C₂²⋊C₄	—	2	2	no	—
72	C₇×C₂²⋊C₄	—	2	1	no	—
73	C₇×C₂²⋊C₄	D₄	1	2	yes	—
74	C₄²×C₇	—	2	2	no	—
75	C₄²×C₇	—	2	1	no	—
76	C₄²×C₇	C₂³	1	1	yes	characteristic
77	#555	C₂×C₇	1	1	yes	characteristic
78	D₄×Q₈	C₂×C₇	1	1	yes	characteristic, complemented
79	#487	C₂×C₇	1	1	yes	characteristic, complemented
80	#485	C₂×C₇	1	2	yes	complemented
81	#477	C₂×C₇	1	1	yes	characteristic, complemented
82	#468	C₂×C₇	1	2	yes	complemented
83	#461	C₂×C₇	1	1	yes	characteristic, complemented
84	#448	C₂×C₇	1	2	yes	complemented
85	C₄×SD₁₆	C₂×C₇	1	1	yes	characteristic, complemented
86	#436	C₂×C₇	1	1	yes	characteristic, complemented
87	#419	C₂×C₇	1	2	yes	complemented
88	C₂³×C₇	—	2	1	no	—
89	C₂³×C₇	C₂×D₄	1	2	yes	—
90	C₇×Q₈	—	2	1	no	—
91	C₇×Q₈	—	2	2	no	—
92	C₇×Q₈	—	2	2	no	—
93	C₇×Q₈	—	4	4	no	—
94	C₇×Q₈	—	8	1	no	—
95	C₇×Q₈	#41	1	2	yes	—
96	C₇×D₄	—	2	2	no	—
97	C₇×D₄	—	2	1	no	—
98	C₇×D₄	—	2	1	no	—
99	C₇×D₄	—	8	2	no	—
100	C₂×C₄×C₇	—	2	1	no	—
101	C₂×C₄×C₇	—	2	2	no	—
102	C₂×C₄×C₇	—	2	2	no	—
103	C₂×C₄×C₇	—	2	1	no	—
104	C₂×C₄×C₇	—	2	1	no	—
105	C₂×C₄×C₇	—	2	2	no	—
106	C₂×C₄×C₇	—	2	2	no	—
107	C₂×C₄×C₇	—	4	4	no	—
108	C₂×C₄×C₇	—	4	2	no	—
109	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
113	C₇×C₈	—	2	1	no	—
114	C₇×C₈	—	2	1	no	—
115	C₇×C₈	—	2	2	no	—
116	C₂²×Q₈	C₂²×C₇	1	2	yes	—
117	C₂×SD₁₆	C₂²×C₇	1	1	yes	characteristic
118	C₂×SD₁₆	C₂²×C₇	1	2	yes	—
119	C₂×M₁₆	C₂²×C₇	1	2	yes	—
120	C₄⋊Q₈	—	2	1	no	—
121	C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
122	C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
123	#124	—	2	2	no	—
124	C₂²⋊Q₈	—	2	2	no	—
125	C₂²⋊Q₈	C₂²×C₇	1	2	yes	—
126	#121	—	2	1	no	—
127	#121	C₂²×C₇	1	2	yes	—
128	C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
129	C₄×D₄	—	2	1	no	—
130	C₄×D₄	C₂²×C₇	1	1	yes	characteristic
131	C₄×D₄	C₂²×C₇	1	1	yes	characteristic
132	#117	—	2	1	no	—
133	#117	C₂²×C₇	1	2	yes	—
134	#107	C₂²×C₇	1	2	yes	—
135	#106	C₂²×C₇	1	1	yes	characteristic
136	C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
137	#103	C₂²×C₇	1	1	yes	characteristic
138	#103	C₂²×C₇	1	2	yes	—
139	#103	C₂²×C₇	1	2	yes	—
140	#103	C₂²×C₇	1	2	yes	—
141	#103	C₂²×C₇	1	2	yes	—
142	#102	C₂²×C₇	1	1	yes	characteristic
143	C₂²⋊C₈	C₂²×C₇	1	2	yes	—
144	C₄×C₈	C₂²×C₇	1	1	yes	characteristic
145	C₂²×C₇	—	2	2	no	—
146	C₂²×C₇	—	2	2	no	—
147	C₂²×C₇	—	2	2	no	—
148	C₂²×C₇	—	4	1	no	—
149	C₂²×C₇	—	4	1	no	—
150	C₂²×C₇	—	4	1	no	—
151	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
152	C₄×C₇	—	2	2	no	—
153	C₄×C₇	—	2	2	no	—
154	C₄×C₇	—	2	1	no	—
155	C₄×C₇	—	4	2	no	—
156	C₄×C₇	—	4	1	no	—
157	C₄×C₇	—	4	1	no	—
158	C₄×C₇	—	4	2	no	—
159	C₄×C₇	#137	1	2	yes	—
160	C₄×C₇	C₂×#41	1	1	yes	characteristic
161	C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
162	C₂×Q₈	—	2	2	no	—
163	C₂×Q₈	—	2	1	no	—
164	C₂×Q₈	—	2	4	no	—
165	C₂×Q₈	—	4	2	no	—
166	C₂×Q₈	C₂³×C₇	1	2	yes	—
167	C₂×Q₈	C₂³×C₇	1	1	yes	characteristic
168	C₂×D₄	—	2	2	no	—
169	C₂×D₄	C₇×D₄	1	1	yes	characteristic
170	C₂×D₄	C₂³×C₇	1	1	yes	characteristic
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	C₂³×C₇	1	2	yes	—
174	SD₁₆	—	2	2	no	—
175	SD₁₆	—	4	2	no	—
176	M₁₆	—	4	2	no	—
177	C₂×C₈	C₂³×C₇	1	1	yes	characteristic
178	C₂×C₈	C₂³×C₇	1	1	yes	characteristic
179	C₂×C₈	C₂³×C₇	1	2	yes	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	2	1	no	—
183	C₄⋊C₄	—	2	1	no	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
186	C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
187	C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
188	C₄⋊C₄	C₂³×C₇	1	2	yes	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₂²⋊C₄	—	2	2	no	—
192	C₂²⋊C₄	—	2	1	no	—
193	C₂²⋊C₄	C₇×D₄	1	2	yes	—
194	C₄²	—	2	2	no	—
195	C₄²	—	2	1	no	—
196	C₄²	C₂³×C₇	1	1	yes	characteristic
197	C₂×C₇	—	4	2	no	—
198	C₂×C₇	—	8	1	no	—
199	C₂×C₇	#546	1	1	yes	characteristic
200	C₂×C₇	C₂×#137	1	1	yes	characteristic
201	C₂×C₇	#577	1	1	yes	characteristic
202	C₂³	—	2	1	no	—
203	C₂³	C₂×C₇×D₄	1	2	yes	—
204	Q₈	—	2	1	no	—
205	Q₈	—	2	2	no	—
206	Q₈	—	2	2	no	—
207	Q₈	—	4	4	no	—
208	Q₈	—	8	1	no	—
209	Q₈	C₇×#41	1	2	yes	—
210	D₄	—	2	2	no	—
211	D₄	—	2	1	no	—
212	D₄	—	2	1	no	—
213	D₄	—	8	2	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
224	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
225	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
226	C₂×C₄	C₂⁴×C₇	1	1	yes	characteristic
227	C₈	—	2	1	no	—
228	C₈	—	2	1	no	—
229	C₈	—	2	2	no	—
230	C₇	#3329	1	1	yes	characteristic, direct product factor, Sylow
231	C₂²	—	2	2	no	—
232	C₂²	—	2	2	no	—
233	C₂²	—	2	2	no	—
234	C₂²	—	4	1	no	—
235	C₂²	—	4	1	no	—
236	C₂²	—	4	1	no	—
237	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
238	C₄	—	2	2	no	—
239	C₄	—	2	2	no	—
240	C₄	—	2	1	no	—
241	C₄	—	4	2	no	—
242	C₄	—	4	1	no	—
243	C₄	—	4	2	no	—
244	C₄	—	4	1	no	—
245	C₄	C₇×#137	1	2	yes	—
246	C₄	C₂×C₇×#41	1	1	yes	characteristic
247	C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
248	C₂	—	4	2	no	—
249	C₂	—	8	1	no	—
250	C₂	C₇×#546	1	1	yes	characteristic
251	C₂	C₂×C₇×#137	1	1	yes	characteristic
252	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	C₇×#555	C₂
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
2. C₇×D₄×Q₈	C₂	semidirect
3. C₇×#487	C₂	semidirect
4. C₇×#485	C₂	semidirect
5. C₇×#477	C₂	semidirect
6. C₇×#468	C₂	semidirect
7. C₇×#461	C₂	semidirect
8. C₇×#448	C₂	semidirect
9. C₄×C₇×SD₁₆	C₂	semidirect
10. C₇×#436	C₂	semidirect
11. C₇×#419	C₂	semidirect
41. #3329	C₇	direct (N × H)
78. D₄×Q₈	C₂×C₇	semidirect
79. #487	C₂×C₇	semidirect
80. #485	C₂×C₇	semidirect
81. #477	C₂×C₇	semidirect
82. #468	C₂×C₇	semidirect
83. #461	C₂×C₇	semidirect
84. #448	C₂×C₇	semidirect
85. C₄×SD₁₆	C₂×C₇	semidirect
86. #436	C₂×C₇	semidirect
87. #419	C₂×C₇	semidirect
230. C₇	#3329	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×#555	2
C₇×D₄×Q₈	2
C₇×#487	2
C₇×#485	2
C₇×#477	2
C₇×#468	2
C₇×#461	2
C₇×#448	2
C₄×C₇×SD₁₆	2
C₇×#436	2
C₇×#419	2
#3329	7

Subgroups — C7×#3329 #11742638

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#3329 #11742638