Subgroups — C₇×#20336 #13197552

Order 1792.

2162 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×#131	C₂	1	2	yes	maximal
2	C₂×C₄×C₇×#41	C₂	1	1	yes	maximal, characteristic
3	C₇×#1729	C₂	1	6	yes	maximal, complemented
4	C₇×#1728	C₂	1	3	yes	maximal, complemented
5	C₂×C₇×#336	C₂	1	3	yes	maximal, complemented
6	C₂²×C₇×#41	C₂²	1	1	yes	characteristic
7	C₂×C₇×#131	—	2	8	no	—
8	C₂×C₇×#131	—	2	8	no	—
9	C₂×C₇×#131	C₄	1	16	yes	complemented
10	C₂²×C₇×M₁₆	C₂²	1	6	yes	—
11	C₂³×C₇×C₈	C₂²	1	6	yes	—
12	C₄×C₇×#41	—	2	8	no	—
13	C₂×C₄×C₇×Q₈	C₄	1	2	yes	—
14	C₂×C₄×C₇×D₄	C₄	1	6	yes	—
15	C₂×C₇×#117	C₂²	1	3	yes	—
16	C₂²×C₄²×C₇	C₂²	1	3	yes	—
17	C₇×#336	C₂²	1	1	yes	characteristic
18	C₇×#336	C₂²	1	3	yes	—
19	C₇×#336	C₂²	1	6	yes	—
20	C₇×#336	C₂²	1	6	yes	—
21	#20336	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂×C₇×#41	C₂×C₄	1	8	yes	—
23	C₂×C₇×#41	D₄	1	8	yes	—
24	C₂×C₇×#41	D₄	1	4	yes	—
25	C₂×C₇×#41	Q₈	1	4	yes	—
26	C₂²×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₇×D₄	C₂×C₄	1	3	yes	—
28	C₂³×C₄×C₇	C₂×C₄	1	6	yes	—
29	C₂³×C₄×C₇	C₂³	1	3	yes	—
30	C₇×#131	—	2	64	no	—
31	C₂×C₇×M₁₆	—	2	12	no	—
32	C₂×C₇×M₁₆	—	2	12	no	—
33	C₂×C₇×M₁₆	C₂×C₄	1	24	yes	complemented
34	C₂²×C₇×C₈	—	2	12	no	—
35	C₂²×C₇×C₈	—	2	12	no	—
36	C₂²×C₇×C₈	C₂×C₄	1	24	yes	complemented
37	C₂²×C₇×C₈	C₂³	1	2	yes	—
38	C₂²×C₇×C₈	C₂³	1	6	yes	—
39	C₄×C₇×Q₈	—	2	8	no	—
40	C₄×C₇×D₄	—	2	24	no	—
41	C₇×#117	—	2	12	no	—
42	C₂×C₇×C₄⋊C₄	C₂×C₄	1	6	yes	—
43	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	6	yes	—
44	C₂×C₄²×C₇	—	2	12	no	—
45	C₂×C₄²×C₇	C₂×C₄	1	6	yes	—
46	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₇	C₂³	1	3	yes	—
48	C₂²×#131	C₂×C₇	1	2	yes	—
49	C₂×C₄×#41	C₂×C₇	1	1	yes	characteristic
50	#1729	C₂×C₇	1	6	yes	complemented
51	#1728	C₂×C₇	1	3	yes	complemented
52	C₂×#336	C₂×C₇	1	3	yes	complemented
53	C₂⁴×C₇	C₂²×C₄	1	3	yes	—
54	C₇×#41	C₂²⋊C₄	1	32	yes	—
55	C₇×#41	C₄⋊C₄	1	32	yes	—
56	C₂×C₇×Q₈	C₄²	1	4	yes	—
57	C₂×C₇×Q₈	C₂²⋊C₄	1	4	yes	—
58	C₂×C₇×Q₈	C₄⋊C₄	1	4	yes	—
59	C₂×C₇×D₄	C₄²	1	12	yes	—
60	C₂×C₇×D₄	C₂²⋊C₄	1	12	yes	—
61	C₂×C₇×D₄	C₄⋊C₄	1	12	yes	—
62	C₂²×C₄×C₇	—	2	12	no	—
63	C₂²×C₄×C₇	—	2	12	no	—
64	C₂²×C₄×C₇	—	4	6	no	—
65	C₂²×C₄×C₇	C₂²×C₄	1	6	yes	—
66	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
68	C₂²×C₄×C₇	C₂²×C₄	1	12	yes	—
69	C₂²×C₄×C₇	C₂×D₄	1	6	yes	—
70	C₂²×C₄×C₇	C₂×D₄	1	12	yes	—
71	C₂²×C₄×C₇	C₂×Q₈	1	6	yes	—
72	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
73	C₇×M₁₆	—	2	48	no	—
74	C₂×C₇×C₈	—	2	48	no	—
75	C₂×C₇×C₈	—	2	2	no	—
76	C₂×C₇×C₈	—	2	2	no	—
77	C₂×C₇×C₈	—	2	6	no	—
78	C₂×C₇×C₈	—	2	6	no	—
79	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
80	C₂×C₇×C₈	C₂²×C₄	1	12	yes	—
81	C₇×C₄⋊C₄	—	2	12	no	—
82	C₇×C₂²⋊C₄	—	2	12	no	—
83	C₄²×C₇	—	2	12	no	—
84	C₄²×C₇	—	2	2	no	—
85	C₄²×C₇	—	2	6	no	—
86	C₂²×#41	C₂²×C₇	1	1	yes	characteristic
87	C₂×#131	—	2	8	no	—
88	C₂×#131	—	2	8	no	—
89	C₂×#131	C₄×C₇	1	16	yes	complemented
90	C₂²×M₁₆	C₂²×C₇	1	6	yes	—
91	C₂³×C₈	C₂²×C₇	1	6	yes	—
92	C₄×#41	—	2	8	no	—
93	C₂×C₄×Q₈	C₄×C₇	1	2	yes	—
94	C₂×C₄×D₄	C₄×C₇	1	6	yes	—
95	C₂×#117	C₂²×C₇	1	3	yes	—
96	C₂²×C₄²	C₂²×C₇	1	3	yes	—
97	#336	C₂²×C₇	1	1	yes	characteristic
98	#336	C₂²×C₇	1	3	yes	—
99	#336	C₂²×C₇	1	6	yes	—
100	#336	C₂²×C₇	1	6	yes	—
101	C₂³×C₇	—	2	6	no	—
102	C₂³×C₇	—	2	6	no	—
103	C₂³×C₇	C₂×C₄²	1	6	yes	—
104	C₂³×C₇	C₂×C₂²⋊C₄	1	6	yes	—
105	C₂³×C₇	C₂×C₄⋊C₄	1	6	yes	—
106	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
107	C₇×Q₈	#95	1	16	yes	—
108	C₇×D₄	#95	1	48	yes	—
109	C₂×C₄×C₇	—	2	2	no	—
110	C₂×C₄×C₇	—	2	2	no	—
111	C₂×C₄×C₇	—	2	6	no	—
112	C₂×C₄×C₇	—	2	6	no	—
113	C₂×C₄×C₇	—	2	6	no	—
114	C₂×C₄×C₇	—	4	24	no	—
115	C₂×C₄×C₇	C₂×C₄²	1	6	yes	—
116	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	24	yes	—
117	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	6	yes	—
118	C₂×C₄×C₇	C₂×C₄⋊C₄	1	24	yes	—
119	C₂×C₄×C₇	C₂×C₄⋊C₄	1	6	yes	—
120	C₂×C₄×C₇	#131	1	4	yes	complemented
121	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
122	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
124	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
125	C₇×C₈	—	2	4	no	—
126	C₇×C₈	—	2	12	no	—
127	C₂×#41	C₂×C₄×C₇	1	8	yes	—
128	C₂×#41	C₇×D₄	1	8	yes	—
129	C₂×#41	C₇×D₄	1	4	yes	—
130	C₂×#41	C₇×Q₈	1	4	yes	—
131	C₂²×Q₈	C₂×C₄×C₇	1	1	yes	characteristic
132	C₂²×D₄	C₂×C₄×C₇	1	3	yes	—
133	C₂³×C₄	C₂×C₄×C₇	1	6	yes	—
134	C₂³×C₄	C₂³×C₇	1	3	yes	—
135	#131	—	2	64	no	—
136	C₂×M₁₆	—	2	12	no	—
137	C₂×M₁₆	—	2	12	no	—
138	C₂×M₁₆	C₂×C₄×C₇	1	24	yes	complemented
139	C₂²×C₈	—	2	12	no	—
140	C₂²×C₈	—	2	12	no	—
141	C₂²×C₈	C₂×C₄×C₇	1	24	yes	complemented
142	C₂²×C₈	C₂³×C₇	1	2	yes	—
143	C₂²×C₈	C₂³×C₇	1	6	yes	—
144	C₄×Q₈	—	2	8	no	—
145	C₄×D₄	—	2	24	no	—
146	#117	—	2	12	no	—
147	C₂×C₄⋊C₄	C₂×C₄×C₇	1	6	yes	—
148	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	6	yes	—
149	C₂×C₄²	—	2	12	no	—
150	C₂×C₄²	C₂×C₄×C₇	1	6	yes	—
151	C₂×C₄²	C₂³×C₇	1	3	yes	—
152	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
153	C₂²×C₇	—	2	6	no	—
154	C₂²×C₇	—	2	6	no	—
155	C₂²×C₇	—	2	12	no	—
156	C₂²×C₇	—	2	12	no	—
157	C₂²×C₇	C₂×#95	1	12	yes	—
158	C₂²×C₇	C₂²×C₄²	1	1	yes	characteristic
159	C₂²×C₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
160	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
161	C₂²×C₇	C₂×#131	1	2	yes	—
162	C₂²×C₇	#568	1	2	yes	—
163	C₄×C₇	—	2	4	no	—
164	C₄×C₇	—	4	6	no	—
165	C₄×C₇	C₂×#95	1	12	yes	—
166	C₄×C₇	C₂²×C₂²⋊C₄	1	2	yes	—
167	C₄×C₇	C₂²×C₄⋊C₄	1	2	yes	—
168	C₂⁴	C₂²×C₄×C₇	1	3	yes	—
169	#41	C₇×C₂²⋊C₄	1	32	yes	—
170	#41	C₇×C₄⋊C₄	1	32	yes	—
171	C₂×Q₈	C₄²×C₇	1	4	yes	—
172	C₂×Q₈	C₇×C₂²⋊C₄	1	4	yes	—
173	C₂×Q₈	C₇×C₄⋊C₄	1	4	yes	—
174	C₂×D₄	C₄²×C₇	1	12	yes	—
175	C₂×D₄	C₇×C₂²⋊C₄	1	12	yes	—
176	C₂×D₄	C₇×C₄⋊C₄	1	12	yes	—
177	C₂²×C₄	—	2	12	no	—
178	C₂²×C₄	—	2	12	no	—
179	C₂²×C₄	—	4	6	no	—
180	C₂²×C₄	C₂²×C₄×C₇	1	6	yes	—
181	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
182	C₂²×C₄	C₂²×C₄×C₇	1	3	yes	—
183	C₂²×C₄	C₂²×C₄×C₇	1	12	yes	—
184	C₂²×C₄	C₂×C₇×D₄	1	6	yes	—
185	C₂²×C₄	C₂×C₇×D₄	1	12	yes	—
186	C₂²×C₄	C₂×C₇×Q₈	1	6	yes	—
187	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
188	M₁₆	—	2	48	no	—
189	C₂×C₈	—	2	48	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	6	no	—
193	C₂×C₈	—	2	6	no	—
194	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
195	C₂×C₈	C₂²×C₄×C₇	1	12	yes	—
196	C₄⋊C₄	—	2	12	no	—
197	C₂²⋊C₄	—	2	12	no	—
198	C₄²	—	2	12	no	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	6	no	—
201	C₂×C₇	—	2	12	no	—
202	C₂×C₇	C₂²×#95	1	1	yes	characteristic
203	C₂×C₇	C₄×#131	1	1	yes	characteristic
204	C₂×C₇	#2875	1	1	yes	characteristic
205	C₂×C₇	#2880	1	2	yes	—
206	C₂×C₇	#2906	1	2	yes	—
207	C₂³	—	2	6	no	—
208	C₂³	—	2	6	no	—
209	C₂³	C₂×C₄²×C₇	1	6	yes	—
210	C₂³	C₂×C₇×C₂²⋊C₄	1	6	yes	—
211	C₂³	C₂×C₇×C₄⋊C₄	1	6	yes	—
212	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
213	Q₈	C₇×#95	1	16	yes	—
214	D₄	C₇×#95	1	48	yes	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	6	no	—
218	C₂×C₄	—	2	6	no	—
219	C₂×C₄	—	2	6	no	—
220	C₂×C₄	—	4	24	no	—
221	C₂×C₄	C₂×C₄²×C₇	1	6	yes	—
222	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	6	yes	—
223	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	24	yes	—
224	C₂×C₄	C₂×C₇×C₄⋊C₄	1	6	yes	—
225	C₂×C₄	C₂×C₇×C₄⋊C₄	1	24	yes	—
226	C₂×C₄	C₇×#131	1	4	yes	complemented
227	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
228	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
229	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
230	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
231	C₈	—	2	4	no	—
232	C₈	—	2	12	no	—
233	C₇	#20336	1	1	yes	characteristic, direct product factor, Sylow
234	C₂²	—	2	12	no	—
235	C₂²	—	2	6	no	—
236	C₂²	—	2	6	no	—
237	C₂²	—	2	12	no	—
238	C₂²	C₂×C₇×#95	1	12	yes	—
239	C₂²	C₂²×C₄²×C₇	1	1	yes	characteristic
240	C₂²	C₂²×C₇×C₂²⋊C₄	1	1	yes	characteristic
241	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
242	C₂²	C₂×C₇×#131	1	2	yes	—
243	C₂²	C₇×#568	1	2	yes	—
244	C₄	—	2	4	no	—
245	C₄	—	4	6	no	—
246	C₄	C₂×C₇×#95	1	12	yes	—
247	C₄	C₂²×C₇×C₂²⋊C₄	1	2	yes	—
248	C₄	C₂²×C₇×C₄⋊C₄	1	2	yes	—
249	C₂	—	2	12	no	—
250	C₂	C₂²×C₇×#95	1	1	yes	characteristic
251	C₂	C₄×C₇×#131	1	1	yes	characteristic
252	C₂	C₇×#2875	1	1	yes	characteristic
253	C₂	C₇×#2880	1	2	yes	—
254	C₂	C₇×#2906	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₇	C₂⁴
commutator	C₂²	C₂²×C₄²×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂³×C₇	C₂³×C₄
prime radical	C₂²×C₇×#41	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#1729	C₂	semidirect
4. C₇×#1728	C₂	semidirect
5. C₂×C₇×#336	C₂	semidirect
9. C₂×C₇×#131	C₄	semidirect
21. #20336	C₇	direct (N × H)
33. C₂×C₇×M₁₆	C₂×C₄	semidirect
36. C₂²×C₇×C₈	C₂×C₄	semidirect
50. #1729	C₂×C₇	semidirect
51. #1728	C₂×C₇	semidirect
52. C₂×#336	C₂×C₇	semidirect
89. C₂×#131	C₄×C₇	semidirect
120. C₂×C₄×C₇	#131	semidirect
138. C₂×M₁₆	C₂×C₄×C₇	semidirect
141. C₂²×C₈	C₂×C₄×C₇	semidirect
226. C₂×C₄	C₇×#131	semidirect
233. C₇	#20336	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₇×#131	2
C₂×C₄×C₇×#41	2
C₇×#1729	2
C₇×#1728	2
C₂×C₇×#336	2
#20336	7

Subgroups — C7×#20336 #13197552

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#20336 #13197552