Subgroups — C₇×#33421 #13210637

Order 1792.

582 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×#131	C₂	1	1	yes	maximal, characteristic
2	C₇×#2164	C₂	1	4	yes	maximal, complemented
3	C₇×#2163	C₂	1	4	yes	maximal, complemented
4	C₇×#2151	C₂	1	2	yes	maximal, complemented
5	C₇×#2113	C₂	1	2	yes	maximal
6	C₄×C₇×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₇×#346	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₇×#131	C₄	1	1	yes	characteristic
9	C₂×C₇×#131	C₂²	1	1	yes	characteristic
10	C₄×C₇×#41	C₄	1	1	yes	characteristic
11	C₂×C₇×M₃₂	C₂²	1	2	yes	—
12	C₂²×C₇×C₁₆	C₂²	1	2	yes	—
13	C₇×#405	C₄	1	2	yes	—
14	C₇×#405	C₄	1	1	yes	characteristic
15	C₇×#405	C₂²	1	1	yes	characteristic
16	C₇×#405	C₂²	1	2	yes	—
17	C₄×C₇×M₁₆	C₄	1	1	yes	characteristic
18	C₄×C₇×M₁₆	C₄	1	2	yes	—
19	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
21	C₇×C₄⋊C₁₆	C₂²	1	8	yes	—
22	C₇×C₂²⋊C₁₆	C₂²	1	8	yes	—
23	C₇×#346	C₂²	1	2	yes	—
24	C₇×#346	C₂²	1	2	yes	—
25	C₇×#346	C₂²	1	2	yes	—
26	C₄×C₇×C₁₆	C₂²	1	2	yes	—
27	#33421	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂×C₇×#41	C₂×C₄	1	1	yes	characteristic
29	C₇×#131	—	2	4	no	—
30	C₇×#131	—	2	4	no	—
31	C₂×C₇×M₁₆	C₂×C₄	1	1	yes	characteristic
32	C₂×C₇×M₁₆	C₂×C₄	1	2	yes	—
33	C₂×C₇×M₁₆	C₂×C₄	1	1	yes	characteristic
34	C₂×C₇×M₁₆	C₂×C₄	1	2	yes	—
35	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
36	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₇×C₈	C₂³	1	2	yes	—
38	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
39	C₄×C₇×Q₈	C₈	1	2	yes	—
40	C₄×C₇×D₄	C₈	1	4	yes	—
41	C₄×C₇×D₄	C₈	1	2	yes	—
42	C₇×#117	C₂×C₄	1	2	yes	—
43	C₇×#117	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
46	C₇×M₃₂	—	2	4	no	—
47	C₂×C₇×C₁₆	—	2	4	no	—
48	C₂×C₇×C₁₆	C₂³	1	4	yes	—
49	C₂×C₇×C₁₆	C₂³	1	4	yes	—
50	C₇×#97	C₂×C₄	1	2	yes	—
51	C₇×#97	C₂×C₄	1	2	yes	—
52	C₇×#97	C₂×C₄	1	2	yes	—
53	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
54	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
55	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
56	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
57	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
58	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
59	C₄×C₇×C₈	C₂³	1	2	yes	—
60	C₄×#131	C₂×C₇	1	1	yes	characteristic
61	#2164	C₂×C₇	1	4	yes	complemented
62	#2163	C₂×C₇	1	4	yes	complemented
63	#2151	C₂×C₇	1	2	yes	complemented
64	#2113	C₂×C₇	1	2	yes	—
65	C₄×M₃₂	C₂×C₇	1	1	yes	characteristic, complemented
66	C₂×#346	C₂×C₇	1	1	yes	characteristic, complemented
67	C₇×#41	—	2	2	no	—
68	C₇×#41	—	2	2	no	—
69	C₂×C₇×Q₈	C₂×C₈	1	1	yes	characteristic
70	C₂×C₇×D₄	C₂×C₈	1	2	yes	—
71	C₂×C₇×D₄	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₄×C₇	C₂×C₈	1	4	yes	—
73	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
74	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
76	C₇×M₁₆	—	2	2	no	—
77	C₇×M₁₆	—	2	2	no	—
78	C₇×M₁₆	—	2	4	no	—
79	C₇×M₁₆	—	2	4	no	—
80	C₂×C₇×C₈	—	2	4	no	—
81	C₂×C₇×C₈	—	2	4	no	—
82	C₂×C₇×C₈	M₁₆	1	4	yes	—
83	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
84	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
85	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₇×C₈	#41	1	4	yes	—
89	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
90	C₇×C₄⋊C₄	C₂×C₈	1	4	yes	—
91	C₇×C₄⋊C₄	C₂×C₈	1	2	yes	—
92	C₇×C₂²⋊C₄	C₂×C₈	1	4	yes	—
93	C₇×C₂²⋊C₄	C₂×C₈	1	2	yes	—
94	C₄²×C₇	C₂×C₈	1	4	yes	—
95	C₄²×C₇	C₂×C₈	1	2	yes	—
96	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₇	C₂²×C₄	1	2	yes	—
99	C₇×C₁₆	—	2	4	no	—
100	C₇×C₁₆	—	2	4	no	—
101	C₂×#131	C₄×C₇	1	1	yes	characteristic
102	C₂×#131	C₂²×C₇	1	1	yes	characteristic
103	C₄×#41	C₄×C₇	1	1	yes	characteristic
104	C₂×M₃₂	C₂²×C₇	1	2	yes	—
105	C₂²×C₁₆	C₂²×C₇	1	2	yes	—
106	#405	C₄×C₇	1	2	yes	—
107	#405	C₄×C₇	1	1	yes	characteristic
108	#405	C₂²×C₇	1	1	yes	characteristic
109	#405	C₂²×C₇	1	2	yes	—
110	C₄×M₁₆	C₄×C₇	1	1	yes	characteristic
111	C₄×M₁₆	C₄×C₇	1	2	yes	—
112	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
113	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
114	C₄⋊C₁₆	C₂²×C₇	1	8	yes	—
115	C₂²⋊C₁₆	C₂²×C₇	1	8	yes	—
116	#346	C₂²×C₇	1	2	yes	—
117	#346	C₂²×C₇	1	2	yes	—
118	#346	C₂²×C₇	1	2	yes	—
119	C₄×C₁₆	C₂²×C₇	1	2	yes	—
120	C₂³×C₇	C₂²×C₈	1	2	yes	—
121	C₂³×C₇	C₂²×C₈	1	1	yes	characteristic
122	C₇×Q₈	—	2	2	no	—
123	C₇×D₄	—	2	4	no	—
124	C₇×D₄	—	2	2	no	—
125	C₂×C₄×C₇	—	2	2	no	—
126	C₂×C₄×C₇	—	2	2	no	—
127	C₂×C₄×C₇	—	4	2	no	—
128	C₂×C₄×C₇	—	4	4	no	—
129	C₂×C₄×C₇	#117	1	2	yes	—
130	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
131	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
132	C₂×C₄×C₇	C₂²×C₈	1	4	yes	—
133	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
134	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
135	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
136	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
137	C₇×C₈	—	2	2	no	—
138	C₇×C₈	—	2	2	no	—
139	C₇×C₈	#117	1	2	yes	—
140	C₇×C₈	C₂×M₁₆	1	2	yes	—
141	C₇×C₈	C₂×M₁₆	1	2	yes	—
142	C₇×C₈	C₂×#41	1	2	yes	—
143	C₂×#41	C₂×C₄×C₇	1	1	yes	characteristic
144	#131	—	2	4	no	—
145	#131	—	2	4	no	—
146	C₂×M₁₆	C₂×C₄×C₇	1	1	yes	characteristic
147	C₂×M₁₆	C₂×C₄×C₇	1	2	yes	—
148	C₂×M₁₆	C₂×C₄×C₇	1	1	yes	characteristic
149	C₂×M₁₆	C₂×C₄×C₇	1	2	yes	—
150	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
151	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
152	C₂²×C₈	C₂³×C₇	1	2	yes	—
153	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
154	C₄×Q₈	C₇×C₈	1	2	yes	—
155	C₄×D₄	C₇×C₈	1	4	yes	—
156	C₄×D₄	C₇×C₈	1	2	yes	—
157	#117	C₂×C₄×C₇	1	2	yes	—
158	#117	C₂×C₄×C₇	1	1	yes	characteristic
159	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
160	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
161	M₃₂	—	2	4	no	—
162	C₂×C₁₆	—	2	4	no	—
163	C₂×C₁₆	C₂³×C₇	1	4	yes	—
164	C₂×C₁₆	C₂³×C₇	1	4	yes	—
165	#97	C₂×C₄×C₇	1	2	yes	—
166	#97	C₂×C₄×C₇	1	2	yes	—
167	#97	C₂×C₄×C₇	1	2	yes	—
168	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
169	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
170	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
171	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
172	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
173	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
174	C₄×C₈	C₂³×C₇	1	2	yes	—
175	C₂²×C₇	—	2	2	no	—
176	C₂²×C₇	—	2	4	no	—
177	C₂²×C₇	—	2	2	no	—
178	C₂²×C₇	#431	1	2	yes	—
179	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
180	C₄×C₇	—	2	2	no	—
181	C₄×C₇	—	2	2	no	—
182	C₄×C₇	—	2	4	no	—
183	C₄×C₇	—	2	2	no	—
184	C₄×C₇	#431	1	2	yes	—
185	C₄×C₇	C₂×#117	1	1	yes	characteristic
186	C₄×C₇	C₂²×M₁₆	1	1	yes	characteristic
187	#41	—	2	2	no	—
188	#41	—	2	2	no	—
189	C₂×Q₈	C₂×C₇×C₈	1	1	yes	characteristic
190	C₂×D₄	C₂×C₇×C₈	1	2	yes	—
191	C₂×D₄	C₂×C₇×C₈	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₇×C₈	1	4	yes	—
193	C₂²×C₄	C₂×C₇×C₈	1	2	yes	—
194	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
195	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
196	M₁₆	—	2	2	no	—
197	M₁₆	—	2	2	no	—
198	M₁₆	—	2	4	no	—
199	M₁₆	—	2	4	no	—
200	C₂×C₈	—	2	4	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	C₇×M₁₆	1	4	yes	—
203	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
204	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
205	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
206	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
207	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
208	C₂×C₈	C₇×#41	1	4	yes	—
209	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
210	C₄⋊C₄	C₂×C₇×C₈	1	4	yes	—
211	C₄⋊C₄	C₂×C₇×C₈	1	2	yes	—
212	C₂²⋊C₄	C₂×C₇×C₈	1	4	yes	—
213	C₂²⋊C₄	C₂×C₇×C₈	1	2	yes	—
214	C₄²	C₂×C₇×C₈	1	4	yes	—
215	C₄²	C₂×C₇×C₈	1	2	yes	—
216	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
217	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
218	C₄²	C₂²×C₄×C₇	1	2	yes	—
219	C₁₆	—	2	4	no	—
220	C₁₆	—	2	4	no	—
221	C₂×C₇	—	2	2	no	—
222	C₂×C₇	—	4	2	no	—
223	C₂×C₇	C₂×#431	1	1	yes	characteristic
224	C₂×C₇	#3407	1	2	yes	—
225	C₂³	C₂²×C₇×C₈	1	2	yes	—
226	C₂³	C₂²×C₇×C₈	1	1	yes	characteristic
227	Q₈	—	2	2	no	—
228	D₄	—	2	4	no	—
229	D₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	C₇×#117	1	2	yes	—
235	C₂×C₄	C₂²×C₇×C₈	1	2	yes	—
236	C₂×C₄	C₂²×C₇×C₈	1	2	yes	—
237	C₂×C₄	C₂²×C₇×C₈	1	4	yes	—
238	C₂×C₄	C₂²×C₇×C₈	1	2	yes	—
239	C₂×C₄	C₂²×C₇×C₈	1	1	yes	characteristic
240	C₂×C₄	C₂×C₇×M₁₆	1	2	yes	—
241	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
242	C₈	—	2	2	no	—
243	C₈	—	2	2	no	—
244	C₈	C₇×#117	1	2	yes	—
245	C₈	C₂×C₇×M₁₆	1	2	yes	—
246	C₈	C₂×C₇×M₁₆	1	2	yes	—
247	C₈	C₂×C₇×#41	1	2	yes	—
248	C₇	#33421	1	1	yes	characteristic, direct product factor, Sylow
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	4	no	—
252	C₂²	C₇×#431	1	2	yes	—
253	C₂²	C₂³×C₇×C₈	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	2	4	no	—
258	C₄	C₇×#431	1	2	yes	—
259	C₄	C₂×C₇×#117	1	1	yes	characteristic
260	C₄	C₂²×C₇×M₁₆	1	1	yes	characteristic
261	C₂	—	2	2	no	—
262	C₂	—	4	2	no	—
263	C₂	C₂×C₇×#431	1	1	yes	characteristic
264	C₂	C₇×#3407	1	2	yes	—

	subgroup	G/H
center	C₂×C₇×C₈	C₂⁴
commutator	C₂²	C₂³×C₇×C₈
Frattini, cosocle	C₂×C₈	C₂⁴×C₇
socle	C₂²×C₇	C₂³×C₈
prime radical	C₂×C₇×#41	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#2164	C₂	semidirect
3. C₇×#2163	C₂	semidirect
4. C₇×#2151	C₂	semidirect
6. C₄×C₇×M₃₂	C₂	semidirect
7. C₂×C₇×#346	C₂	semidirect
27. #33421	C₇	direct (N × H)
61. #2164	C₂×C₇	semidirect
62. #2163	C₂×C₇	semidirect
63. #2151	C₂×C₇	semidirect
65. C₄×M₃₂	C₂×C₇	semidirect
66. C₂×#346	C₂×C₇	semidirect
248. C₇	#33421	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×#131	2
C₇×#2164	2
C₇×#2163	2
C₇×#2151	2
C₇×#2113	2
C₄×C₇×M₃₂	2
C₂×C₇×#346	2
#33421	7

Subgroups — C7×#33421 #13210637

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#33421 #13210637