Subgroups — C₇×#17975 #13195191

Order 1792.

726 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3169	C₂	1	1	yes	maximal, characteristic
2	C₇×#2931	C₂	1	2	yes	maximal, complemented
3	C₇×#1581	C₂	1	8	yes	maximal, complemented
4	C₇×#1567	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1492	C₂	1	2	yes	maximal, complemented
6	C₂×C₇×#322	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₇×#131	—	2	2	no	—
8	C₇×#532	C₂²	1	2	yes	—
9	C₇×#500	C₄	1	1	yes	characteristic
10	C₇×#493	C₄	1	1	yes	characteristic
11	C₇×#486	C₄	1	2	yes	—
12	C₇×#466	C₄	1	4	yes	—
13	C₄×C₇×D₈	C₂²	1	4	yes	—
14	C₇×#436	—	2	2	no	—
15	C₇×#436	C₂²	1	4	yes	—
16	C₇×C₈×D₄	—	2	2	no	—
17	C₇×C₈×D₄	C₂²	1	4	yes	—
18	C₇×#423	C₂²	1	2	yes	—
19	C₇×#408	—	2	2	no	—
20	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
23	C₇×#335	C₂²	1	4	yes	—
24	C₇×D₄⋊C₈	C₂²	1	8	yes	—
25	C₇×#322	C₂²	1	4	yes	—
26	#17975	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂×C₇×#41	—	2	2	no	—
28	C₂×C₇×D₈	C₂×C₄	1	4	yes	—
29	C₇×#131	—	4	8	no	—
30	C₂×C₇×M₁₆	—	2	2	no	—
31	C₂×C₇×M₁₆	—	2	4	no	—
32	C₂²×C₇×C₈	—	2	1	no	—
33	C₂²×C₇×C₈	—	2	4	no	—
34	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
35	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
36	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
37	C₇×C₄⋊Q₈	C₂×C₄	1	2	yes	—
38	C₇×#127	C₂×C₄	1	2	yes	—
39	C₇×#124	—	2	2	no	—
40	C₇×#121	—	2	2	no	—
41	C₇×#121	C₂×C₄	1	4	yes	—
42	C₄×C₇×D₄	—	2	2	no	—
43	C₄×C₇×D₄	C₂³	1	4	yes	—
44	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
45	C₇×#107	C₂×C₄	1	4	yes	—
46	C₇×C₄⋊C₈	—	2	2	no	—
47	C₇×C₄⋊C₈	C₂³	1	4	yes	—
48	C₇×#102	C₂×C₄	1	8	yes	—
49	C₇×C₂²⋊C₈	—	2	4	no	—
50	C₇×C₂²⋊C₈	—	2	4	no	—
51	C₄×C₇×C₈	—	2	1	no	—
52	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
53	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
54	C₄×C₇×C₈	D₄	1	2	yes	—
55	C₄×C₇×C₈	C₂³	1	2	yes	—
56	C₄×C₇×C₈	C₂³	1	2	yes	—
57	C₄×C₇×C₈	C₂³	1	2	yes	—
58	#3169	C₂×C₇	1	1	yes	characteristic
59	#2931	C₂×C₇	1	2	yes	complemented
60	#1581	C₂×C₇	1	8	yes	complemented
61	#1567	C₂×C₇	1	1	yes	characteristic, complemented
62	#1492	C₂×C₇	1	2	yes	complemented
63	C₂×#322	C₂×C₇	1	1	yes	characteristic, complemented
64	C₇×#41	—	4	4	no	—
65	C₇×#41	—	4	4	no	—
66	C₂×C₇×Q₈	—	2	2	no	—
67	C₂×C₇×D₄	—	2	4	no	—
68	C₂×C₇×D₄	—	2	2	no	—
69	C₂×C₇×D₄	—	2	2	no	—
70	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₇	—	2	4	no	—
72	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₇	#41	1	1	yes	characteristic
75	C₇×D₈	—	4	4	no	—
76	C₇×M₁₆	—	4	4	no	—
77	C₇×M₁₆	—	4	8	no	—
78	C₂×C₇×C₈	—	2	2	no	—
79	C₂×C₇×C₈	—	2	4	no	—
80	C₂×C₇×C₈	—	2	2	no	—
81	C₂×C₇×C₈	—	4	2	no	—
82	C₂×C₇×C₈	—	4	2	no	—
83	C₂×C₇×C₈	—	4	8	no	—
84	C₂×C₇×C₈	D₈	1	4	yes	complemented
85	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
86	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
87	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
88	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
89	C₇×C₄⋊C₄	—	2	2	no	—
90	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
91	C₇×C₂²⋊C₄	—	2	4	no	—
92	C₇×C₂²⋊C₄	—	2	4	no	—
93	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₇	#41	1	1	yes	characteristic
96	C₄²×C₇	C₂⁴	1	1	yes	characteristic
97	C₂×#131	—	2	2	no	—
98	#532	C₂²×C₇	1	2	yes	—
99	#500	C₄×C₇	1	1	yes	characteristic
100	#493	C₄×C₇	1	1	yes	characteristic
101	#486	C₄×C₇	1	2	yes	—
102	#466	C₄×C₇	1	4	yes	—
103	C₄×D₈	C₂²×C₇	1	4	yes	—
104	#436	—	2	2	no	—
105	#436	C₂²×C₇	1	4	yes	—
106	C₈×D₄	—	2	2	no	—
107	C₈×D₄	C₂²×C₇	1	4	yes	—
108	#423	C₂²×C₇	1	2	yes	—
109	#408	—	2	2	no	—
110	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
111	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
112	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
113	#335	C₂²×C₇	1	4	yes	—
114	D₄⋊C₈	C₂²×C₇	1	8	yes	—
115	#322	C₂²×C₇	1	4	yes	—
116	C₂³×C₇	—	2	4	no	—
117	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
118	C₇×Q₈	—	4	4	no	—
119	C₇×D₄	—	2	8	no	—
120	C₇×D₄	—	4	8	no	—
121	C₇×D₄	—	4	4	no	—
122	C₇×D₄	—	4	4	no	—
123	C₂×C₄×C₇	—	2	2	no	—
124	C₂×C₄×C₇	—	2	2	no	—
125	C₂×C₄×C₇	—	2	4	no	—
126	C₂×C₄×C₇	—	4	4	no	—
127	C₂×C₄×C₇	—	4	4	no	—
128	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
132	C₂×C₄×C₇	#135	1	2	yes	—
133	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
136	C₇×C₈	—	2	2	no	—
137	C₇×C₈	—	2	2	no	—
138	C₇×C₈	—	2	2	no	—
139	C₇×C₈	—	4	2	no	—
140	C₇×C₈	—	4	4	no	—
141	C₇×C₈	C₂×D₈	1	4	yes	—
142	C₂×#41	—	2	2	no	—
143	C₂×D₈	C₂×C₄×C₇	1	4	yes	—
144	#131	—	4	8	no	—
145	C₂×M₁₆	—	2	2	no	—
146	C₂×M₁₆	—	2	4	no	—
147	C₂²×C₈	—	2	1	no	—
148	C₂²×C₈	—	2	4	no	—
149	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
150	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
151	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
152	C₄⋊Q₈	C₂×C₄×C₇	1	2	yes	—
153	#127	C₂×C₄×C₇	1	2	yes	—
154	#124	—	2	2	no	—
155	#121	—	2	2	no	—
156	#121	C₂×C₄×C₇	1	4	yes	—
157	C₄×D₄	—	2	2	no	—
158	C₄×D₄	C₂³×C₇	1	4	yes	—
159	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
160	#107	C₂×C₄×C₇	1	4	yes	—
161	C₄⋊C₈	—	2	2	no	—
162	C₄⋊C₈	C₂³×C₇	1	4	yes	—
163	#102	C₂×C₄×C₇	1	8	yes	—
164	C₂²⋊C₈	—	2	4	no	—
165	C₂²⋊C₈	—	2	4	no	—
166	C₄×C₈	—	2	1	no	—
167	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
168	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
169	C₄×C₈	C₇×D₄	1	2	yes	—
170	C₄×C₈	C₂³×C₇	1	2	yes	—
171	C₄×C₈	C₂³×C₇	1	2	yes	—
172	C₄×C₈	C₂³×C₇	1	2	yes	—
173	C₂²×C₇	—	2	1	no	—
174	C₂²×C₇	—	2	1	no	—
175	C₂²×C₇	—	4	4	no	—
176	C₂²×C₇	—	4	4	no	—
177	C₂²×C₇	—	4	4	no	—
178	C₂²×C₇	C₄×D₈	1	2	yes	—
179	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
180	C₄×C₇	—	2	1	no	—
181	C₄×C₇	—	2	1	no	—
182	C₄×C₇	—	4	4	no	—
183	C₄×C₇	C₄×D₈	1	2	yes	—
184	C₄×C₇	#444	1	2	yes	—
185	C₄×C₇	#568	1	2	yes	—
186	C₄×C₇	C₂²×D₈	1	1	yes	characteristic
187	C₄×C₇	C₂×#135	1	1	yes	characteristic
188	#41	—	4	4	no	—
189	#41	—	4	4	no	—
190	C₂×Q₈	—	2	2	no	—
191	C₂×D₄	—	2	4	no	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
198	C₂²×C₄	C₇×#41	1	1	yes	characteristic
199	D₈	—	4	4	no	—
200	M₁₆	—	4	4	no	—
201	M₁₆	—	4	8	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	4	2	no	—
207	C₂×C₈	—	4	8	no	—
208	C₂×C₈	C₇×D₈	1	4	yes	complemented
209	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
210	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
211	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
212	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
215	C₂²⋊C₄	—	2	4	no	—
216	C₂²⋊C₄	—	2	4	no	—
217	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
218	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
219	C₄²	C₇×#41	1	1	yes	characteristic
220	C₄²	C₂⁴×C₇	1	1	yes	characteristic
221	C₂×C₇	—	2	2	no	—
222	C₂×C₇	—	8	4	no	—
223	C₂×C₇	#2932	1	1	yes	characteristic
224	C₂×C₇	C₂×C₄×D₈	1	1	yes	characteristic
225	C₂×C₇	C₂×#444	1	1	yes	characteristic
226	C₂³	—	2	4	no	—
227	C₂³	C₄×C₇×D₄	1	1	yes	characteristic
228	Q₈	—	4	4	no	—
229	D₄	—	2	8	no	—
230	D₄	—	4	8	no	—
231	D₄	—	4	4	no	—
232	D₄	—	4	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
239	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
240	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
241	C₂×C₄	C₂×C₇×D₈	1	2	yes	—
242	C₂×C₄	C₇×#135	1	2	yes	—
243	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
244	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
245	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
246	C₈	—	2	2	no	—
247	C₈	—	2	2	no	—
248	C₈	—	2	2	no	—
249	C₈	—	4	2	no	—
250	C₈	—	4	4	no	—
251	C₈	C₂×C₇×D₈	1	4	yes	—
252	C₇	#17975	1	1	yes	characteristic, direct product factor, Sylow
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	4	4	no	—
256	C₂²	—	4	4	no	—
257	C₂²	—	4	4	no	—
258	C₂²	C₄×C₇×D₈	1	2	yes	—
259	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	4	4	no	—
263	C₄	C₄×C₇×D₈	1	2	yes	—
264	C₄	C₇×#444	1	2	yes	—
265	C₄	C₇×#568	1	2	yes	—
266	C₄	C₂²×C₇×D₈	1	1	yes	characteristic
267	C₄	C₂×C₇×#135	1	1	yes	characteristic
268	C₂	—	2	2	no	—
269	C₂	—	8	4	no	—
270	C₂	C₇×#2932	1	1	yes	characteristic
271	C₂	C₂×C₄×C₇×D₈	1	1	yes	characteristic
272	C₂	C₂×C₇×#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₄²	C₂⁴×C₇
socle	C₂²×C₇	C₂×C₄×D₄
prime radical	C₇×#3169	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#2931	C₂	semidirect
3. C₇×#1581	C₂	semidirect
4. C₇×#1567	C₂	semidirect
5. C₇×#1492	C₂	semidirect
6. C₂×C₇×#322	C₂	semidirect
26. #17975	C₇	direct (N × H)
59. #2931	C₂×C₇	semidirect
60. #1581	C₂×C₇	semidirect
61. #1567	C₂×C₇	semidirect
62. #1492	C₂×C₇	semidirect
63. C₂×#322	C₂×C₇	semidirect
84. C₂×C₇×C₈	D₈	semidirect
208. C₂×C₈	C₇×D₈	semidirect
252. C₇	#17975	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×#3169	2
C₇×#2931	2
C₇×#1581	2
C₇×#1567	2
C₇×#1492	2
C₂×C₇×#322	2
#17975	7

Subgroups — C7×#17975 #13195191

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#17975 #13195191