Subgroups — C₇×#23196 #13200412

Order 1792.

884 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3147	C₂	1	1	yes	maximal, characteristic
2	C₇×#2957	C₂	1	4	yes	maximal, complemented
3	C₇×#1976	C₂	1	4	yes	maximal, complemented
4	C₇×#1884	C₂	1	4	yes	maximal, complemented
5	C₇×#1772	C₂	1	2	yes	maximal, complemented
6	C₂×C₇×#135	C₂²	1	4	yes	—
7	C₇×#568	C₂²	1	4	yes	—
8	C₇×#532	C₂²	1	2	yes	—
9	C₇×#495	C₄	1	4	yes	complemented
10	C₇×#494	C₄	1	1	yes	characteristic, complemented
11	C₇×#493	C₄	1	1	yes	characteristic, complemented
12	C₇×#492	C₄	1	2	yes	complemented
13	C₇×#444	—	2	8	no	—
14	C₇×#443	—	2	8	no	—
15	C₂×C₇×#108	C₂²	1	2	yes	—
16	C₇×#427	C₂²	1	2	yes	—
17	C₇×#421	C₂²	1	8	yes	—
18	C₇×#410	C₂²	1	4	yes	—
19	C₇×#405	C₂²	1	4	yes	—
20	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
21	C₇×#339	C₂²	1	4	yes	—
22	#23196	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂×C₇×#41	C₂³	1	4	yes	—
24	C₇×#135	—	2	4	no	—
25	C₇×#135	—	2	4	no	—
26	C₇×#135	—	2	4	no	—
27	C₇×#135	—	2	4	no	—
28	C₂×C₇×Q₁₆	C₂×C₄	1	4	yes	—
29	C₂×C₇×SD₁₆	C₂×C₄	1	8	yes	—
30	C₂×C₇×D₈	C₂×C₄	1	4	yes	—
31	C₇×#131	—	2	16	no	—
32	C₂×C₇×M₁₆	—	4	4	no	—
33	C₂×C₇×M₁₆	C₂³	1	4	yes	—
34	C₂²×C₇×C₈	C₂³	1	2	yes	—
35	C₇×C₄⋊Q₈	C₂×C₄	1	2	yes	—
36	C₇×#127	C₂×C₄	1	2	yes	—
37	C₇×#124	C₂×C₄	1	4	yes	—
38	C₇×#121	—	4	2	no	—
39	C₄×C₇×D₄	—	4	2	no	—
40	C₇×#117	C₂³	1	4	yes	—
41	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
42	C₇×#108	D₄	1	4	yes	complemented
43	C₇×#108	D₄	1	4	yes	complemented
44	C₇×#107	—	2	2	no	—
45	C₇×#106	—	2	2	no	—
46	C₇×#104	—	2	16	no	—
47	C₇×#99	—	4	4	no	—
48	C₇×#97	—	2	4	no	—
49	C₄×C₇×C₈	—	2	4	no	—
50	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
51	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
52	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
53	#3147	C₂×C₇	1	1	yes	characteristic
54	#2957	C₂×C₇	1	4	yes	complemented
55	#1976	C₂×C₇	1	4	yes	complemented
56	#1884	C₂×C₇	1	4	yes	complemented
57	#1772	C₂×C₇	1	2	yes	complemented
58	C₇×#41	—	2	8	no	—
59	C₇×#41	—	2	8	no	—
60	C₂×C₇×Q₈	C₂²×C₄	1	4	yes	—
61	C₂×C₇×D₄	—	2	4	no	—
62	C₂×C₇×D₄	—	4	2	no	—
63	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₇	—	2	1	no	—
65	C₂²×C₄×C₇	—	2	4	no	—
66	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
67	C₇×Q₁₆	—	2	4	no	—
68	C₇×Q₁₆	—	2	4	no	—
69	C₇×SD₁₆	—	2	8	no	—
70	C₇×SD₁₆	—	2	8	no	—
71	C₇×D₈	—	2	4	no	—
72	C₇×D₈	—	2	4	no	—
73	C₇×M₁₆	—	2	4	no	—
74	C₇×M₁₆	—	4	8	no	—
75	C₇×M₁₆	C₂×D₄	1	8	yes	—
76	C₂×C₇×C₈	—	2	4	no	—
77	C₂×C₇×C₈	—	4	8	no	—
78	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
79	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
80	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
81	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
82	C₂×C₇×C₈	#41	1	2	yes	—
83	C₂×C₇×C₈	#41	1	2	yes	—
84	C₇×C₄⋊C₄	—	2	4	no	—
85	C₇×C₄⋊C₄	—	4	2	no	—
86	C₇×C₂²⋊C₄	—	2	4	no	—
87	C₇×C₂²⋊C₄	—	4	4	no	—
88	C₄²×C₇	—	2	1	no	—
89	C₄²×C₇	—	2	4	no	—
90	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
92	C₂×#135	C₂²×C₇	1	4	yes	—
93	#568	C₂²×C₇	1	4	yes	—
94	#532	C₂²×C₇	1	2	yes	—
95	#495	C₄×C₇	1	4	yes	complemented
96	#494	C₄×C₇	1	1	yes	characteristic, complemented
97	#493	C₄×C₇	1	1	yes	characteristic, complemented
98	#492	C₄×C₇	1	2	yes	complemented
99	#444	—	2	8	no	—
100	#443	—	2	8	no	—
101	C₂×#108	C₂²×C₇	1	2	yes	—
102	#427	C₂²×C₇	1	2	yes	—
103	#421	C₂²×C₇	1	8	yes	—
104	#410	C₂²×C₇	1	4	yes	—
105	#405	C₂²×C₇	1	4	yes	—
106	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
107	#339	C₂²×C₇	1	4	yes	—
108	C₂³×C₇	—	2	4	no	—
109	C₂³×C₇	H(5,𝔽₂)	1	1	yes	characteristic
110	C₇×Q₈	—	2	8	no	—
111	C₇×D₄	—	2	8	no	—
112	C₇×D₄	—	4	8	no	—
113	C₇×D₄	—	8	4	no	—
114	C₂×C₄×C₇	—	2	1	no	—
115	C₂×C₄×C₇	—	2	1	no	—
116	C₂×C₄×C₇	—	2	4	no	—
117	C₂×C₄×C₇	—	4	2	no	—
118	C₂×C₄×C₇	—	4	4	no	—
119	C₂×C₄×C₇	—	4	4	no	—
120	C₂×C₄×C₇	—	4	4	no	—
121	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
123	C₂×C₄×C₇	C₂×#41	1	2	yes	—
124	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
125	C₇×C₈	—	2	8	no	—
126	C₇×C₈	C₄×D₄	1	2	yes	—
127	C₇×C₈	C₄×D₄	1	2	yes	—
128	C₇×C₈	C₄×D₄	1	2	yes	—
129	C₇×C₈	C₄×D₄	1	2	yes	—
130	C₂×#41	C₂³×C₇	1	4	yes	—
131	#135	—	2	4	no	—
132	#135	—	2	4	no	—
133	#135	—	2	4	no	—
134	#135	—	2	4	no	—
135	C₂×Q₁₆	C₂×C₄×C₇	1	4	yes	—
136	C₂×SD₁₆	C₂×C₄×C₇	1	8	yes	—
137	C₂×D₈	C₂×C₄×C₇	1	4	yes	—
138	#131	—	2	16	no	—
139	C₂×M₁₆	—	4	4	no	—
140	C₂×M₁₆	C₂³×C₇	1	4	yes	—
141	C₂²×C₈	C₂³×C₇	1	2	yes	—
142	C₄⋊Q₈	C₂×C₄×C₇	1	2	yes	—
143	#127	C₂×C₄×C₇	1	2	yes	—
144	#124	C₂×C₄×C₇	1	4	yes	—
145	#121	—	4	2	no	—
146	C₄×D₄	—	4	2	no	—
147	#117	C₂³×C₇	1	4	yes	—
148	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
149	#108	C₇×D₄	1	4	yes	complemented
150	#108	C₇×D₄	1	4	yes	complemented
151	#107	—	2	2	no	—
152	#106	—	2	2	no	—
153	#104	—	2	16	no	—
154	#99	—	4	4	no	—
155	#97	—	2	4	no	—
156	C₄×C₈	—	2	4	no	—
157	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
158	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
159	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
160	C₂²×C₇	—	4	1	no	—
161	C₂²×C₇	—	4	4	no	—
162	C₂²×C₇	—	8	4	no	—
163	C₂²×C₇	#518	1	1	yes	characteristic
164	C₂²×C₇	#550	1	2	yes	—
165	C₄×C₇	—	2	2	no	—
166	C₄×C₇	—	4	1	no	—
167	C₄×C₇	—	4	4	no	—
168	C₄×C₇	—	8	4	no	—
169	C₄×C₇	C₂×C₄×D₄	1	2	yes	—
170	C₄×C₇	D₄²	1	1	yes	characteristic
171	C₄×C₇	#559	1	1	yes	characteristic
172	#41	—	2	8	no	—
173	#41	—	2	8	no	—
174	C₂×Q₈	C₂²×C₄×C₇	1	4	yes	—
175	C₂×D₄	—	2	4	no	—
176	C₂×D₄	—	4	2	no	—
177	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
181	Q₁₆	—	2	4	no	—
182	Q₁₆	—	2	4	no	—
183	SD₁₆	—	2	8	no	—
184	SD₁₆	—	2	8	no	—
185	D₈	—	2	4	no	—
186	D₈	—	2	4	no	—
187	M₁₆	—	2	4	no	—
188	M₁₆	—	4	8	no	—
189	M₁₆	C₂×C₇×D₄	1	8	yes	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	4	8	no	—
192	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
193	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
194	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
195	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
196	C₂×C₈	C₇×#41	1	2	yes	—
197	C₂×C₈	C₇×#41	1	2	yes	—
198	C₄⋊C₄	—	2	4	no	—
199	C₄⋊C₄	—	4	2	no	—
200	C₂²⋊C₄	—	2	4	no	—
201	C₂²⋊C₄	—	4	4	no	—
202	C₄²	—	2	1	no	—
203	C₄²	—	2	4	no	—
204	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
205	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
206	C₂×C₇	—	2	1	no	—
207	C₂×C₇	—	2	2	no	—
208	C₂×C₇	—	8	4	no	—
209	C₂×C₇	#2366	1	1	yes	characteristic
210	C₂³	—	2	4	no	—
211	C₂³	C₇×H(5,𝔽₂)	1	1	yes	characteristic
212	Q₈	—	2	8	no	—
213	D₄	—	2	8	no	—
214	D₄	—	4	8	no	—
215	D₄	—	8	4	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
224	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
225	C₂×C₄	C₂×C₇×#41	1	2	yes	—
226	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
227	C₈	—	2	8	no	—
228	C₈	C₄×C₇×D₄	1	2	yes	—
229	C₈	C₄×C₇×D₄	1	2	yes	—
230	C₈	C₄×C₇×D₄	1	2	yes	—
231	C₈	C₄×C₇×D₄	1	2	yes	—
232	C₇	#23196	1	1	yes	characteristic, direct product factor, Sylow
233	C₂²	—	4	1	no	—
234	C₂²	—	4	4	no	—
235	C₂²	—	8	4	no	—
236	C₂²	C₇×#518	1	1	yes	characteristic
237	C₂²	C₇×#550	1	2	yes	—
238	C₄	—	2	2	no	—
239	C₄	—	4	1	no	—
240	C₄	—	4	4	no	—
241	C₄	—	8	4	no	—
242	C₄	C₂×C₄×C₇×D₄	1	2	yes	—
243	C₄	C₇×D₄²	1	1	yes	characteristic
244	C₄	C₇×#559	1	1	yes	characteristic
245	C₂	—	2	1	no	—
246	C₂	—	2	2	no	—
247	C₂	—	8	4	no	—
248	C₂	C₇×#2366	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₇	D₄²
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂×C₇	#2366
prime radical	C₇×#3147	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂²×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#2957	C₂	semidirect
3. C₇×#1976	C₂	semidirect
4. C₇×#1884	C₂	semidirect
5. C₇×#1772	C₂	semidirect
9. C₇×#495	C₄	semidirect
10. C₇×#494	C₄	semidirect
11. C₇×#493	C₄	semidirect
12. C₇×#492	C₄	semidirect
22. #23196	C₇	direct (N × H)
42. C₇×#108	D₄	semidirect
43. C₇×#108	D₄	semidirect
54. #2957	C₂×C₇	semidirect
55. #1976	C₂×C₇	semidirect
56. #1884	C₂×C₇	semidirect
57. #1772	C₂×C₇	semidirect
95. #495	C₄×C₇	semidirect
96. #494	C₄×C₇	semidirect
97. #493	C₄×C₇	semidirect
98. #492	C₄×C₇	semidirect
149. #108	C₇×D₄	semidirect
150. #108	C₇×D₄	semidirect
232. C₇	#23196	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×#3147	2
C₇×#2957	2
C₇×#1976	2
C₇×#1884	2
C₇×#1772	2
#23196	7

Subgroups — C7×#23196 #13200412

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#23196 #13200412