Subgroups — C₇×#33706 #13210922

Order 1792.

598 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3160	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#495	C₂	1	1	yes	maximal, characteristic
3	C₇×#2243	C₂	1	4	yes	maximal, complemented
4	C₇×#2242	C₂	1	4	yes	maximal, complemented
5	C₇×#2141	C₂	1	2	yes	maximal, complemented
6	C₇×#2140	C₂	1	2	yes	maximal
7	C₄×C₇×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₇×Q₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₇×SD₁₆	—	2	1	no	—
10	C₂²×C₇×D₈	C₂²	1	1	yes	characteristic
11	C₇×#533	—	2	1	no	—
12	C₂×C₇×#124	—	2	1	no	—
13	C₂×C₇×M₃₂	C₂²	1	2	yes	—
14	C₇×#500	C₂²	1	2	yes	complemented
15	C₇×#499	—	2	1	no	—
16	C₇×#499	C₂²	1	2	yes	complemented
17	C₇×#498	—	2	1	no	—
18	C₇×#495	—	2	1	no	—
19	C₇×#495	—	2	1	no	—
20	C₇×#495	C₂²	1	2	yes	—
21	C₇×#495	C₂²	1	2	yes	—
22	C₇×#427	—	2	1	no	—
23	C₇×#427	C₂²	1	2	yes	—
24	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
25	C₇×#358	C₂²	1	8	yes	complemented
26	C₇×#357	C₂²	1	8	yes	—
27	C₇×#346	C₂²	1	2	yes	complemented
28	C₄×C₇×C₁₆	C₂²	1	2	yes	complemented
29	#33706	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂²×C₇×Q₈	—	2	1	no	—
31	C₂²×C₇×D₄	—	2	1	no	—
32	C₂×C₇×Q₁₆	—	2	1	no	—
33	C₂×C₇×Q₁₆	—	2	4	no	—
34	C₂×C₇×Q₁₆	C₂³	1	2	yes	—
35	C₂×C₇×SD₁₆	—	2	2	no	—
36	C₂×C₇×SD₁₆	—	2	2	no	—
37	C₂×C₇×SD₁₆	—	4	4	no	—
38	C₂×C₇×D₈	—	2	1	no	—
39	C₂×C₇×D₈	—	2	4	no	—
40	C₂×C₇×D₈	C₂³	1	2	yes	—
41	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
42	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
43	C₇×C₄⋊Q₈	—	2	2	no	—
44	C₇×#125	—	2	2	no	—
45	C₇×#124	—	2	2	no	—
46	C₇×#124	—	2	2	no	—
47	C₇×#124	—	4	2	no	—
48	C₇×C₂²⋊Q₈	—	4	2	no	—
49	C₄×C₇×Q₈	—	4	2	no	—
50	C₇×#117	—	2	2	no	—
51	C₂×C₇×C₂²⋊C₄	—	4	2	no	—
52	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
53	C₇×M₃₂	—	2	4	no	—
54	C₂×C₇×C₁₆	C₂³	1	4	yes	—
55	C₇×#107	—	2	2	no	—
56	C₇×#107	C₂³	1	4	yes	—
57	C₇×#106	—	2	2	no	—
58	C₇×#106	—	2	2	no	—
59	C₄×C₇×C₈	D₄	1	1	yes	characteristic
60	C₄×C₇×C₈	D₄	1	1	yes	characteristic
61	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
62	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
63	#3160	C₂×C₇	1	1	yes	characteristic, complemented
64	C₂×#495	C₂×C₇	1	1	yes	characteristic
65	#2243	C₂×C₇	1	4	yes	complemented
66	#2242	C₂×C₇	1	4	yes	complemented
67	#2141	C₂×C₇	1	2	yes	complemented
68	#2140	C₂×C₇	1	2	yes	—
69	C₄×M₃₂	C₂×C₇	1	1	yes	characteristic, complemented
70	C₂⁴×C₇	—	4	1	no	—
71	C₂×C₇×Q₈	—	2	2	no	—
72	C₂×C₇×Q₈	—	4	2	no	—
73	C₂×C₇×Q₈	—	4	1	no	—
74	C₂×C₇×Q₈	—	4	2	no	—
75	C₂×C₇×Q₈	—	4	2	no	—
76	C₂×C₇×D₄	—	2	2	no	—
77	C₂×C₇×D₄	—	4	2	no	—
78	C₂×C₇×D₄	—	4	1	no	—
79	C₂×C₇×D₄	—	4	2	no	—
80	C₂²×C₄×C₇	—	4	1	no	—
81	C₂²×C₄×C₇	D₈	1	2	yes	—
82	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
83	C₇×Q₁₆	—	2	4	no	—
84	C₇×Q₁₆	—	4	2	no	—
85	C₇×SD₁₆	—	4	4	no	—
86	C₇×SD₁₆	—	4	4	no	—
87	C₇×D₈	—	2	4	no	—
88	C₇×D₈	—	4	2	no	—
89	C₂×C₇×C₈	SD₁₆	1	4	yes	—
90	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
91	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
92	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
93	C₂×C₇×C₈	#41	1	4	yes	—
94	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
95	C₇×C₄⋊C₄	—	2	4	no	—
96	C₇×C₄⋊C₄	—	4	4	no	—
97	C₇×C₄⋊C₄	—	4	2	no	—
98	C₇×C₂²⋊C₄	—	4	2	no	—
99	C₇×C₂²⋊C₄	—	4	4	no	—
100	C₇×C₂²⋊C₄	—	4	4	no	—
101	C₄²×C₇	—	4	2	no	—
102	C₄²×C₇	D₈	1	2	yes	—
103	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
105	C₇×C₁₆	—	2	4	no	—
106	C₂²×Q₁₆	C₂²×C₇	1	1	yes	characteristic
107	C₂²×SD₁₆	—	2	1	no	—
108	C₂²×D₈	C₂²×C₇	1	1	yes	characteristic
109	#533	—	2	1	no	—
110	C₂×#124	—	2	1	no	—
111	C₂×M₃₂	C₂²×C₇	1	2	yes	—
112	#500	C₂²×C₇	1	2	yes	complemented
113	#499	—	2	1	no	—
114	#499	C₂²×C₇	1	2	yes	complemented
115	#498	—	2	1	no	—
116	#495	—	2	1	no	—
117	#495	—	2	1	no	—
118	#495	C₂²×C₇	1	2	yes	—
119	#495	C₂²×C₇	1	2	yes	—
120	#427	—	2	1	no	—
121	#427	C₂²×C₇	1	2	yes	—
122	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
123	#358	C₂²×C₇	1	8	yes	complemented
124	#357	C₂²×C₇	1	8	yes	—
125	#346	C₂²×C₇	1	2	yes	complemented
126	C₄×C₁₆	C₂²×C₇	1	2	yes	complemented
127	C₂³×C₇	—	4	2	no	—
128	C₂³×C₇	—	8	4	no	—
129	C₂³×C₇	—	8	2	no	—
130	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
131	C₇×Q₈	—	4	2	no	—
132	C₇×Q₈	—	4	2	no	—
133	C₇×Q₈	—	8	4	no	—
134	C₇×Q₈	—	8	2	no	—
135	C₇×D₄	—	4	2	no	—
136	C₇×D₄	—	4	2	no	—
137	C₇×D₄	—	8	2	no	—
138	C₂×C₄×C₇	—	4	2	no	—
139	C₂×C₄×C₇	—	4	4	no	—
140	C₂×C₄×C₇	—	4	2	no	—
141	C₂×C₄×C₇	—	8	2	no	—
142	C₂×C₄×C₇	#124	1	2	yes	—
143	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
144	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
145	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
147	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
148	C₇×C₈	#124	1	2	yes	—
149	C₇×C₈	C₂×SD₁₆	1	2	yes	—
150	C₇×C₈	C₂×SD₁₆	1	2	yes	—
151	C₇×C₈	C₂×#41	1	2	yes	—
152	C₂²×Q₈	—	2	1	no	—
153	C₂²×D₄	—	2	1	no	—
154	C₂×Q₁₆	—	2	1	no	—
155	C₂×Q₁₆	—	2	4	no	—
156	C₂×Q₁₆	C₂³×C₇	1	2	yes	—
157	C₂×SD₁₆	—	2	2	no	—
158	C₂×SD₁₆	—	2	2	no	—
159	C₂×SD₁₆	—	4	4	no	—
160	C₂×D₈	—	2	1	no	—
161	C₂×D₈	—	2	4	no	—
162	C₂×D₈	C₂³×C₇	1	2	yes	—
163	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
164	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
165	C₄⋊Q₈	—	2	2	no	—
166	#125	—	2	2	no	—
167	#124	—	2	2	no	—
168	#124	—	2	2	no	—
169	#124	—	4	2	no	—
170	C₂²⋊Q₈	—	4	2	no	—
171	C₄×Q₈	—	4	2	no	—
172	#117	—	2	2	no	—
173	C₂×C₂²⋊C₄	—	4	2	no	—
174	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
175	M₃₂	—	2	4	no	—
176	C₂×C₁₆	C₂³×C₇	1	4	yes	—
177	#107	—	2	2	no	—
178	#107	C₂³×C₇	1	4	yes	—
179	#106	—	2	2	no	—
180	#106	—	2	2	no	—
181	C₄×C₈	C₇×D₄	1	1	yes	characteristic
182	C₄×C₈	C₇×D₄	1	1	yes	characteristic
183	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
184	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
185	C₂²×C₇	—	2	2	no	—
186	C₂²×C₇	—	8	2	no	—
187	C₂²×C₇	—	8	4	no	—
188	C₂²×C₇	—	16	4	no	—
189	C₂²×C₇	#486	1	2	yes	—
190	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
191	C₄×C₇	—	2	2	no	—
192	C₄×C₇	—	2	2	no	—
193	C₄×C₇	—	8	2	no	—
194	C₄×C₇	—	8	4	no	—
195	C₄×C₇	#486	1	2	yes	—
196	C₄×C₇	C₂×#124	1	1	yes	characteristic
197	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
198	C₂⁴	—	4	1	no	—
199	C₂×Q₈	—	2	2	no	—
200	C₂×Q₈	—	4	2	no	—
201	C₂×Q₈	—	4	1	no	—
202	C₂×Q₈	—	4	2	no	—
203	C₂×Q₈	—	4	2	no	—
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	—	4	2	no	—
206	C₂×D₄	—	4	1	no	—
207	C₂×D₄	—	4	2	no	—
208	C₂²×C₄	—	4	1	no	—
209	C₂²×C₄	C₇×D₈	1	2	yes	—
210	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
211	Q₁₆	—	2	4	no	—
212	Q₁₆	—	4	2	no	—
213	SD₁₆	—	4	4	no	—
214	SD₁₆	—	4	4	no	—
215	D₈	—	2	4	no	—
216	D₈	—	4	2	no	—
217	C₂×C₈	C₇×SD₁₆	1	4	yes	—
218	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
219	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
220	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
221	C₂×C₈	C₇×#41	1	4	yes	—
222	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
223	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	4	4	no	—
225	C₄⋊C₄	—	4	2	no	—
226	C₂²⋊C₄	—	4	2	no	—
227	C₂²⋊C₄	—	4	4	no	—
228	C₂²⋊C₄	—	4	4	no	—
229	C₄²	—	4	2	no	—
230	C₄²	C₇×D₈	1	2	yes	—
231	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
232	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
233	C₁₆	—	2	4	no	—
234	C₂×C₇	—	2	2	no	—
235	C₂×C₇	—	16	2	no	—
236	C₂×C₇	C₂×#486	1	1	yes	characteristic
237	C₂×C₇	#3414	1	2	yes	—
238	C₂³	—	4	2	no	—
239	C₂³	—	8	4	no	—
240	C₂³	—	8	2	no	—
241	C₂³	C₂×C₇×D₈	1	1	yes	characteristic
242	Q₈	—	4	2	no	—
243	Q₈	—	4	2	no	—
244	Q₈	—	8	4	no	—
245	Q₈	—	8	2	no	—
246	D₄	—	4	2	no	—
247	D₄	—	4	2	no	—
248	D₄	—	8	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	8	2	no	—
253	C₂×C₄	C₇×#124	1	2	yes	—
254	C₂×C₄	C₂×C₇×D₈	1	2	yes	—
255	C₂×C₄	C₂×C₇×D₈	1	2	yes	—
256	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
257	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
258	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
259	C₈	C₇×#124	1	2	yes	—
260	C₈	C₂×C₇×SD₁₆	1	2	yes	—
261	C₈	C₂×C₇×SD₁₆	1	2	yes	—
262	C₈	C₂×C₇×#41	1	2	yes	—
263	C₇	#33706	1	1	yes	characteristic, direct product factor, Sylow
264	C₂²	—	2	2	no	—
265	C₂²	—	8	4	no	—
266	C₂²	—	8	2	no	—
267	C₂²	—	16	4	no	—
268	C₂²	C₇×#486	1	2	yes	—
269	C₂²	C₂²×C₇×D₈	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	8	4	no	—
273	C₄	—	8	2	no	—
274	C₄	C₇×#486	1	2	yes	—
275	C₄	C₂×C₇×#124	1	1	yes	characteristic
276	C₄	C₂²×C₇×SD₁₆	1	1	yes	characteristic
277	C₂	—	2	2	no	—
278	C₂	—	16	2	no	—
279	C₂	C₂×C₇×#486	1	1	yes	characteristic
280	C₂	C₇×#3414	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₇
prime radical	C₂²×C₇×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#3160	C₂	semidirect
3. C₇×#2243	C₂	semidirect
4. C₇×#2242	C₂	semidirect
5. C₇×#2141	C₂	semidirect
7. C₄×C₇×M₃₂	C₂	semidirect
14. C₇×#500	C₂²	semidirect
16. C₇×#499	C₂²	semidirect
25. C₇×#358	C₂²	semidirect
27. C₇×#346	C₂²	semidirect
28. C₄×C₇×C₁₆	C₂²	semidirect
29. #33706	C₇	direct (N × H)
63. #3160	C₂×C₇	semidirect
65. #2243	C₂×C₇	semidirect
66. #2242	C₂×C₇	semidirect
67. #2141	C₂×C₇	semidirect
69. C₄×M₃₂	C₂×C₇	semidirect
112. #500	C₂²×C₇	semidirect
114. #499	C₂²×C₇	semidirect
123. #358	C₂²×C₇	semidirect
125. #346	C₂²×C₇	semidirect
126. C₄×C₁₆	C₂²×C₇	semidirect
263. C₇	#33706	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×#3160	2
C₂×C₇×#495	2
C₇×#2243	2
C₇×#2242	2
C₇×#2141	2
C₇×#2140	2
C₄×C₇×M₃₂	2
#33706	7

Subgroups — C7×#33706 #13210922

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#33706 #13210922