Subgroups — C₇×#33697 #13210913

Order 1792.

598 conjugacy classes of subgroups.

The subgroup lattice has 268 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	C₇×#3147	C₂	1	1	yes	maximal, characteristic
3	C₇×#2242	C₂	1	4	yes	maximal, complemented
4	C₇×#2241	C₂	1	2	yes	maximal, complemented
5	C₇×#2240	C₂	1	2	yes	maximal, complemented
6	C₇×#2138	C₂	1	4	yes	maximal, complemented
7	C₂×C₄×C₇×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₇×#135	—	2	1	no	—
9	C₂×C₇×#135	C₂²	1	2	yes	—
10	C₇×#533	—	2	1	no	—
11	C₇×#532	—	2	1	no	—
12	C₂²×C₇×C₁₆	C₂²	1	2	yes	—
13	C₇×#500	C₂²	1	2	yes	—
14	C₇×#499	—	2	1	no	—
15	C₇×#499	C₂²	1	2	yes	—
16	C₇×#498	—	2	1	no	—
17	C₇×#495	—	2	1	no	—
18	C₇×#495	C₂²	1	2	yes	—
19	C₇×#494	C₂²	1	1	yes	characteristic
20	C₇×#493	C₂²	1	1	yes	characteristic
21	C₇×#492	—	2	1	no	—
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	—
26	C₇×#357	C₂²	1	8	yes	—
27	C₄×C₇×C₁₆	C₂²	1	2	yes	—
28	C₄×C₇×C₁₆	C₂²	1	2	yes	—
29	#33697	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂×C₇×#41	—	2	2	no	—
31	C₇×#135	—	2	8	no	—
32	C₇×#135	—	4	4	no	—
33	C₂×C₇×Q₁₆	—	2	1	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×D₈	—	2	1	no	—
38	C₂×C₇×D₈	C₂³	1	2	yes	—
39	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
40	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
41	C₇×C₄⋊Q₈	—	2	2	no	—
42	C₇×C₄⋊Q₈	—	2	1	no	—
43	C₇×#127	—	2	1	no	—
44	C₇×#125	—	2	2	no	—
45	C₇×#124	—	2	2	no	—
46	C₇×C₂²⋊Q₈	—	4	2	no	—
47	C₇×#121	—	4	2	no	—
48	C₄×C₇×Q₈	—	4	2	no	—
49	C₄×C₇×D₄	—	4	2	no	—
50	C₇×#117	—	2	2	no	—
51	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
52	C₂×C₇×C₁₆	—	2	4	no	—
53	C₂×C₇×C₁₆	C₂³	1	4	yes	—
54	C₇×#107	—	2	2	no	—
55	C₇×#107	C₂³	1	4	yes	—
56	C₇×#106	—	2	2	no	—
57	C₇×#106	—	2	2	no	—
58	C₄×C₇×C₈	D₄	1	1	yes	characteristic
59	C₄×C₇×C₈	D₄	1	1	yes	characteristic
60	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
61	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
62	#3160	C₂×C₇	1	1	yes	characteristic, complemented
63	#3147	C₂×C₇	1	1	yes	characteristic
64	#2242	C₂×C₇	1	4	yes	complemented
65	#2241	C₂×C₇	1	2	yes	complemented
66	#2240	C₂×C₇	1	2	yes	complemented
67	#2138	C₂×C₇	1	4	yes	complemented
68	C₂×C₄×C₁₆	C₂×C₇	1	1	yes	characteristic, complemented
69	C₇×#41	—	4	4	no	—
70	C₇×#41	—	4	4	no	—
71	C₂×C₇×Q₈	—	2	2	no	—
72	C₂×C₇×Q₈	—	4	2	no	—
73	C₂×C₇×D₄	—	2	2	no	—
74	C₂×C₇×D₄	—	4	2	no	—
75	C₂×C₇×D₄	—	4	2	no	—
76	C₂²×C₄×C₇	—	4	2	no	—
77	C₂²×C₄×C₇	D₈	1	2	yes	—
78	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
79	C₇×Q₁₆	—	2	4	no	—
80	C₇×Q₁₆	—	4	2	no	—
81	C₇×SD₁₆	—	4	4	no	—
82	C₇×SD₁₆	—	4	4	no	—
83	C₇×D₈	—	2	4	no	—
84	C₇×D₈	—	4	2	no	—
85	C₂×C₇×C₈	SD₁₆	1	4	yes	—
86	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
87	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
88	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
89	C₂×C₇×C₈	#41	1	4	yes	—
90	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
91	C₇×C₄⋊C₄	—	2	4	no	—
92	C₇×C₄⋊C₄	—	4	2	no	—
93	C₇×C₄⋊C₄	—	4	4	no	—
94	C₇×C₄⋊C₄	—	4	2	no	—
95	C₇×C₂²⋊C₄	—	4	2	no	—
96	C₇×C₂²⋊C₄	—	4	4	no	—
97	C₄²×C₇	—	4	2	no	—
98	C₄²×C₇	D₈	1	2	yes	—
99	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
101	C₇×C₁₆	—	2	4	no	—
102	C₂×#135	—	2	1	no	—
103	C₂×#135	C₂²×C₇	1	2	yes	—
104	#533	—	2	1	no	—
105	#532	—	2	1	no	—
106	C₂²×C₁₆	C₂²×C₇	1	2	yes	—
107	#500	C₂²×C₇	1	2	yes	—
108	#499	—	2	1	no	—
109	#499	C₂²×C₇	1	2	yes	—
110	#498	—	2	1	no	—
111	#495	—	2	1	no	—
112	#495	C₂²×C₇	1	2	yes	—
113	#494	C₂²×C₇	1	1	yes	characteristic
114	#493	C₂²×C₇	1	1	yes	characteristic
115	#492	—	2	1	no	—
116	#427	—	2	1	no	—
117	#427	C₂²×C₇	1	2	yes	—
118	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
119	#358	C₂²×C₇	1	8	yes	—
120	#357	C₂²×C₇	1	8	yes	—
121	C₄×C₁₆	C₂²×C₇	1	2	yes	—
122	C₄×C₁₆	C₂²×C₇	1	2	yes	—
123	C₂³×C₇	—	4	2	no	—
124	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
125	C₇×Q₈	—	4	2	no	—
126	C₇×Q₈	—	4	2	no	—
127	C₇×Q₈	—	8	4	no	—
128	C₇×D₄	—	4	2	no	—
129	C₇×D₄	—	4	2	no	—
130	C₇×D₄	—	8	4	no	—
131	C₇×D₄	—	8	4	no	—
132	C₂×C₄×C₇	—	2	4	no	—
133	C₂×C₄×C₇	—	4	4	no	—
134	C₂×C₄×C₇	—	4	2	no	—
135	C₂×C₄×C₇	—	8	4	no	—
136	C₂×C₄×C₇	#124	1	2	yes	—
137	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
138	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
139	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
141	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
142	C₇×C₈	#124	1	2	yes	—
143	C₇×C₈	C₂×SD₁₆	1	2	yes	—
144	C₇×C₈	C₂×SD₁₆	1	2	yes	—
145	C₇×C₈	C₂×#41	1	2	yes	—
146	C₂×#41	—	2	2	no	—
147	#135	—	2	8	no	—
148	#135	—	4	4	no	—
149	C₂×Q₁₆	—	2	1	no	—
150	C₂×Q₁₆	C₂³×C₇	1	2	yes	—
151	C₂×SD₁₆	—	2	2	no	—
152	C₂×SD₁₆	—	2	2	no	—
153	C₂×D₈	—	2	1	no	—
154	C₂×D₈	C₂³×C₇	1	2	yes	—
155	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
156	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
157	C₄⋊Q₈	—	2	2	no	—
158	C₄⋊Q₈	—	2	1	no	—
159	#127	—	2	1	no	—
160	#125	—	2	2	no	—
161	#124	—	2	2	no	—
162	C₂²⋊Q₈	—	4	2	no	—
163	#121	—	4	2	no	—
164	C₄×Q₈	—	4	2	no	—
165	C₄×D₄	—	4	2	no	—
166	#117	—	2	2	no	—
167	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
168	C₂×C₁₆	—	2	4	no	—
169	C₂×C₁₆	C₂³×C₇	1	4	yes	—
170	#107	—	2	2	no	—
171	#107	C₂³×C₇	1	4	yes	—
172	#106	—	2	2	no	—
173	#106	—	2	2	no	—
174	C₄×C₈	C₇×D₄	1	1	yes	characteristic
175	C₄×C₈	C₇×D₄	1	1	yes	characteristic
176	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
177	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
178	C₂²×C₇	—	2	2	no	—
179	C₂²×C₇	—	8	2	no	—
180	C₂²×C₇	—	8	4	no	—
181	C₂²×C₇	#486	1	2	yes	—
182	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
183	C₄×C₇	—	2	2	no	—
184	C₄×C₇	—	8	2	no	—
185	C₄×C₇	—	8	4	no	—
186	C₄×C₇	#486	1	2	yes	—
187	C₄×C₇	#508	1	4	yes	—
188	C₄×C₇	C₂×#124	1	1	yes	characteristic
189	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
190	#41	—	4	4	no	—
191	#41	—	4	4	no	—
192	C₂×Q₈	—	2	2	no	—
193	C₂×Q₈	—	4	2	no	—
194	C₂×D₄	—	2	2	no	—
195	C₂×D₄	—	4	2	no	—
196	C₂×D₄	—	4	2	no	—
197	C₂²×C₄	—	4	2	no	—
198	C₂²×C₄	C₇×D₈	1	2	yes	—
199	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
200	Q₁₆	—	2	4	no	—
201	Q₁₆	—	4	2	no	—
202	SD₁₆	—	4	4	no	—
203	SD₁₆	—	4	4	no	—
204	D₈	—	2	4	no	—
205	D₈	—	4	2	no	—
206	C₂×C₈	C₇×SD₁₆	1	4	yes	—
207	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
208	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
209	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
210	C₂×C₈	C₇×#41	1	4	yes	—
211	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	4	2	no	—
214	C₄⋊C₄	—	4	4	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₂²⋊C₄	—	4	2	no	—
217	C₂²⋊C₄	—	4	4	no	—
218	C₄²	—	4	2	no	—
219	C₄²	C₇×D₈	1	2	yes	—
220	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
221	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
222	C₁₆	—	2	4	no	—
223	C₂×C₇	—	2	2	no	—
224	C₂×C₇	—	16	2	no	—
225	C₂×C₇	C₂×#486	1	1	yes	characteristic
226	C₂×C₇	C₂×#508	1	2	yes	—
227	C₂³	—	4	2	no	—
228	C₂³	C₂×C₇×D₈	1	1	yes	characteristic
229	Q₈	—	4	2	no	—
230	Q₈	—	4	2	no	—
231	Q₈	—	8	4	no	—
232	D₄	—	4	2	no	—
233	D₄	—	4	2	no	—
234	D₄	—	8	4	no	—
235	D₄	—	8	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	8	4	no	—
240	C₂×C₄	C₇×#124	1	2	yes	—
241	C₂×C₄	C₂×C₇×D₈	1	2	yes	—
242	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
243	C₂×C₄	C₂×C₇×D₈	1	2	yes	—
244	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
245	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
246	C₈	C₇×#124	1	2	yes	—
247	C₈	C₂×C₇×SD₁₆	1	2	yes	—
248	C₈	C₂×C₇×SD₁₆	1	2	yes	—
249	C₈	C₂×C₇×#41	1	2	yes	—
250	C₇	#33697	1	1	yes	characteristic, direct product factor, Sylow
251	C₂²	—	2	2	no	—
252	C₂²	—	8	4	no	—
253	C₂²	—	8	2	no	—
254	C₂²	C₇×#486	1	2	yes	—
255	C₂²	C₂²×C₇×D₈	1	1	yes	characteristic
256	C₄	—	2	2	no	—
257	C₄	—	8	4	no	—
258	C₄	—	8	2	no	—
259	C₄	C₇×#486	1	2	yes	—
260	C₄	C₇×#508	1	4	yes	—
261	C₄	C₂×C₇×#124	1	1	yes	characteristic
262	C₄	C₂²×C₇×SD₁₆	1	1	yes	characteristic
263	C₂	—	2	2	no	—
264	C₂	—	16	2	no	—
265	C₂	C₂×C₇×#486	1	1	yes	characteristic
266	C₂	C₂×C₇×#508	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₇
socle	C₂²×C₇	C₂²×D₈
prime radical	C₇×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×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×#2242	C₂	semidirect
4. C₇×#2241	C₂	semidirect
5. C₇×#2240	C₂	semidirect
6. C₇×#2138	C₂	semidirect
7. C₂×C₄×C₇×C₁₆	C₂	semidirect
29. #33697	C₇	direct (N × H)
62. #3160	C₂×C₇	semidirect
64. #2242	C₂×C₇	semidirect
65. #2241	C₂×C₇	semidirect
66. #2240	C₂×C₇	semidirect
67. #2138	C₂×C₇	semidirect
68. C₂×C₄×C₁₆	C₂×C₇	semidirect
250. C₇	#33697	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
C₇×#3147	2
C₇×#2242	2
C₇×#2241	2
C₇×#2240	2
C₇×#2138	2
C₂×C₄×C₇×C₁₆	2
#33697	7

Subgroups — C7×#33697 #13210913

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#33697 #13210913