Subgroups — C₇×#23187 #13200403

Order 1792.

946 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#494	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#441	C₂	1	2	yes	maximal
3	C₇×#2365	C₂	1	2	yes	maximal
4	C₇×#1981	C₂	1	2	yes	maximal
5	C₇×#1971	C₂	1	2	yes	maximal
6	C₇×#1878	C₂	1	4	yes	maximal
7	C₄×C₇×#106	C₂	1	1	yes	maximal, characteristic
8	C₇×#1770	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₇×Q₁₆	C₂²	1	2	yes	—
10	C₂²×C₇×Q₁₆	C₂²	1	2	yes	—
11	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₂×C₄×C₇×Q₈	C₂²	1	4	yes	—
13	C₇×#494	C₄	1	8	yes	complemented
14	C₇×#441	—	2	8	no	—
15	C₂×C₇×#106	C₂²	1	1	yes	characteristic
16	C₂×C₇×#106	C₂²	1	1	yes	characteristic
17	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₇×#103	C₂²	1	4	yes	—
19	C₂×C₇×#103	C₂²	1	4	yes	—
20	C₂×C₇×#97	C₂²	1	2	yes	—
21	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
22	C₇×#391	C₂²	1	4	yes	—
23	C₇×#389	—	2	2	no	—
24	C₇×#387	—	2	4	no	—
25	C₄×C₇×C₄⋊C₄	—	2	2	no	—
26	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
27	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
28	C₇×#340	C₂²	1	2	yes	—
29	C₇×#336	C₂²	1	2	yes	—
30	#23187	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂²×C₇×Q₈	C₂³	1	4	yes	—
32	C₂×C₇×Q₁₆	—	2	4	no	—
33	C₂×C₇×Q₁₆	—	2	4	no	—
34	C₂×C₇×Q₁₆	—	4	2	no	—
35	C₂×C₇×Q₁₆	—	4	2	no	—
36	C₂×C₇×Q₁₆	C₂×C₄	1	8	yes	—
37	C₂×C₇×Q₁₆	C₂×C₄	1	8	yes	—
38	C₂²×C₇×C₈	C₂³	1	2	yes	—
39	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
40	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
41	C₇×C₄⋊Q₈	—	2	4	no	—
42	C₇×C₄⋊Q₈	C₂×C₄	1	8	yes	—
43	C₄×C₇×Q₈	—	2	8	no	—
44	C₄×C₇×Q₈	—	4	4	no	—
45	C₂×C₇×C₄⋊C₄	—	2	2	no	—
46	C₂×C₇×C₄⋊C₄	—	2	2	no	—
47	C₂×C₇×C₄⋊C₄	—	2	4	no	—
48	C₂×C₇×C₄⋊C₄	D₄	1	2	yes	—
49	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
50	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
51	C₂×C₄²×C₇	—	2	2	no	—
52	C₂×C₄²×C₇	—	2	4	no	—
53	C₂×C₄²×C₇	C₂³	1	2	yes	—
54	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
55	C₇×#106	—	2	2	no	—
56	C₇×#106	D₄	1	4	yes	—
57	C₇×C₄⋊C₈	—	2	2	no	—
58	C₇×#103	—	2	8	no	—
59	C₇×#103	—	4	4	no	—
60	C₇×#97	—	2	4	no	—
61	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
62	C₇×#95	—	2	4	no	—
63	C₇×#95	—	2	2	no	—
64	C₇×#95	—	2	4	no	—
65	C₇×#95	D₄	1	2	yes	—
66	C₂×#494	C₂×C₇	1	1	yes	characteristic
67	C₂×#441	C₂×C₇	1	2	yes	—
68	#2365	C₂×C₇	1	2	yes	—
69	#1981	C₂×C₇	1	2	yes	—
70	#1971	C₂×C₇	1	2	yes	—
71	#1878	C₂×C₇	1	4	yes	—
72	C₄×#106	C₂×C₇	1	1	yes	characteristic
73	#1770	C₂×C₇	1	1	yes	characteristic
74	C₂×C₇×Q₈	—	2	4	no	—
75	C₂×C₇×Q₈	—	2	4	no	—
76	C₂×C₇×Q₈	—	2	4	no	—
77	C₂×C₇×Q₈	—	2	4	no	—
78	C₂×C₇×Q₈	—	2	4	no	—
79	C₂×C₇×Q₈	C₂²×C₄	1	8	yes	—
80	C₂²×C₄×C₇	—	2	4	no	—
81	C₂²×C₄×C₇	—	2	2	no	—
82	C₂²×C₄×C₇	—	2	2	no	—
83	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
84	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
87	C₇×Q₁₆	—	2	16	no	—
88	C₇×Q₁₆	—	4	8	no	—
89	C₂×C₇×C₈	—	2	1	no	—
90	C₂×C₇×C₈	—	2	1	no	—
91	C₂×C₇×C₈	—	2	2	no	—
92	C₂×C₇×C₈	—	2	2	no	—
93	C₂×C₇×C₈	—	2	2	no	—
94	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
95	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
96	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
97	C₂×C₇×C₈	#41	1	2	yes	—
98	C₇×C₄⋊C₄	—	2	8	no	—
99	C₇×C₄⋊C₄	—	2	8	no	—
100	C₇×C₄⋊C₄	—	2	4	no	—
101	C₇×C₄⋊C₄	—	2	2	no	—
102	C₇×C₄⋊C₄	—	2	2	no	—
103	C₇×C₄⋊C₄	—	2	2	no	—
104	C₇×C₄⋊C₄	—	4	8	no	—
105	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
106	C₄²×C₇	—	2	2	no	—
107	C₄²×C₇	—	2	2	no	—
108	C₄²×C₇	—	4	8	no	—
109	C₄²×C₇	—	4	4	no	—
110	C₄²×C₇	C₂²×C₄	1	2	yes	—
111	C₄²×C₇	#41	1	2	yes	—
112	C₂²×Q₁₆	C₂²×C₇	1	2	yes	—
113	C₂²×Q₁₆	C₂²×C₇	1	2	yes	—
114	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
115	C₂×C₄×Q₈	C₂²×C₇	1	4	yes	—
116	#494	C₄×C₇	1	8	yes	complemented
117	#441	—	2	8	no	—
118	C₂×#106	C₂²×C₇	1	1	yes	characteristic
119	C₂×#106	C₂²×C₇	1	1	yes	characteristic
120	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
121	C₂×#103	C₂²×C₇	1	4	yes	—
122	C₂×#103	C₂²×C₇	1	4	yes	—
123	C₂×#97	C₂²×C₇	1	2	yes	—
124	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
125	#391	C₂²×C₇	1	4	yes	—
126	#389	—	2	2	no	—
127	#387	—	2	4	no	—
128	C₄×C₄⋊C₄	—	2	2	no	—
129	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
130	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
131	#340	C₂²×C₇	1	2	yes	—
132	#336	C₂²×C₇	1	2	yes	—
133	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
134	C₇×Q₈	—	2	8	no	—
135	C₇×Q₈	—	2	8	no	—
136	C₇×Q₈	—	4	4	no	—
137	C₇×Q₈	—	4	4	no	—
138	C₂×C₄×C₇	—	2	2	no	—
139	C₂×C₄×C₇	—	2	8	no	—
140	C₂×C₄×C₇	—	2	4	no	—
141	C₂×C₄×C₇	—	2	4	no	—
142	C₂×C₄×C₇	—	2	4	no	—
143	C₂×C₄×C₇	—	2	4	no	—
144	C₂×C₄×C₇	—	4	4	no	—
145	C₂×C₄×C₇	—	4	4	no	—
146	C₂×C₄×C₇	—	4	2	no	—
147	C₂×C₄×C₇	—	4	2	no	—
148	C₂×C₄×C₇	—	4	2	no	—
149	C₂×C₄×C₇	—	4	2	no	—
150	C₂×C₄×C₇	—	4	2	no	—
151	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
152	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
153	C₂×C₄×C₇	#137	1	4	yes	—
154	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
155	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
157	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
158	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
159	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
160	C₇×C₈	—	2	2	no	—
161	C₇×C₈	—	4	2	no	—
162	C₇×C₈	C₄×D₄	1	4	yes	—
163	C₂²×Q₈	C₂³×C₇	1	4	yes	—
164	C₂×Q₁₆	—	2	4	no	—
165	C₂×Q₁₆	—	2	4	no	—
166	C₂×Q₁₆	—	4	2	no	—
167	C₂×Q₁₆	—	4	2	no	—
168	C₂×Q₁₆	C₂×C₄×C₇	1	8	yes	—
169	C₂×Q₁₆	C₂×C₄×C₇	1	8	yes	—
170	C₂²×C₈	C₂³×C₇	1	2	yes	—
171	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
172	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
173	C₄⋊Q₈	—	2	4	no	—
174	C₄⋊Q₈	C₂×C₄×C₇	1	8	yes	—
175	C₄×Q₈	—	2	8	no	—
176	C₄×Q₈	—	4	4	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₄⋊C₄	—	2	4	no	—
180	C₂×C₄⋊C₄	C₇×D₄	1	2	yes	—
181	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
182	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
183	C₂×C₄²	—	2	4	no	—
184	C₂×C₄²	—	2	2	no	—
185	C₂×C₄²	C₂³×C₇	1	2	yes	—
186	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
187	#106	—	2	2	no	—
188	#106	C₇×D₄	1	4	yes	—
189	C₄⋊C₈	—	2	2	no	—
190	#103	—	2	8	no	—
191	#103	—	4	4	no	—
192	#97	—	2	4	no	—
193	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
194	#95	—	2	4	no	—
195	#95	—	2	2	no	—
196	#95	—	2	4	no	—
197	#95	C₇×D₄	1	2	yes	—
198	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
199	C₂²×C₇	D₄²	1	1	yes	characteristic
200	C₂²×C₇	#550	1	1	yes	characteristic
201	C₂²×C₇	C₂×#137	1	2	yes	—
202	C₂²×C₇	#578	1	2	yes	—
203	C₄×C₇	—	2	2	no	—
204	C₄×C₇	—	4	8	no	—
205	C₄×C₇	—	4	4	no	—
206	C₄×C₇	—	4	4	no	—
207	C₄×C₇	—	8	2	no	—
208	C₄×C₇	#441	1	4	yes	complemented
209	C₄×C₇	C₂×C₄×D₄	1	1	yes	characteristic
210	C₄×C₇	#518	1	1	yes	characteristic
211	C₄×C₇	#550	1	1	yes	characteristic
212	C₄×C₇	#559	1	1	yes	characteristic
213	C₂×Q₈	—	2	4	no	—
214	C₂×Q₈	—	2	4	no	—
215	C₂×Q₈	—	2	4	no	—
216	C₂×Q₈	—	2	4	no	—
217	C₂×Q₈	—	2	4	no	—
218	C₂×Q₈	C₂²×C₄×C₇	1	8	yes	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
223	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
225	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
226	Q₁₆	—	2	16	no	—
227	Q₁₆	—	4	8	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
234	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
235	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
236	C₂×C₈	C₇×#41	1	2	yes	—
237	C₄⋊C₄	—	2	8	no	—
238	C₄⋊C₄	—	2	8	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	4	8	no	—
244	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	4	8	no	—
248	C₄²	—	4	4	no	—
249	C₄²	C₂²×C₄×C₇	1	2	yes	—
250	C₄²	C₇×#41	1	2	yes	—
251	C₂×C₇	#2366	1	1	yes	characteristic
252	C₂×C₇	C₂×#441	1	1	yes	characteristic
253	C₂×C₇	#2950	1	1	yes	characteristic
254	C₂×C₇	#3276	1	2	yes	—
255	C₂×C₇	#3398	1	2	yes	—
256	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
257	Q₈	—	2	8	no	—
258	Q₈	—	2	8	no	—
259	Q₈	—	4	4	no	—
260	Q₈	—	4	4	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
275	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
276	C₂×C₄	C₇×#137	1	4	yes	—
277	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
278	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
279	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
280	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
281	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
282	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
283	C₈	—	2	2	no	—
284	C₈	—	4	2	no	—
285	C₈	C₄×C₇×D₄	1	4	yes	—
286	C₇	#23187	1	1	yes	characteristic, direct product factor, Sylow
287	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
288	C₂²	C₇×D₄²	1	1	yes	characteristic
289	C₂²	C₇×#550	1	1	yes	characteristic
290	C₂²	C₂×C₇×#137	1	2	yes	—
291	C₂²	C₇×#578	1	2	yes	—
292	C₄	—	2	2	no	—
293	C₄	—	4	4	no	—
294	C₄	—	4	4	no	—
295	C₄	—	4	8	no	—
296	C₄	—	8	2	no	—
297	C₄	C₇×#441	1	4	yes	complemented
298	C₄	C₂×C₄×C₇×D₄	1	1	yes	characteristic
299	C₄	C₇×#518	1	1	yes	characteristic
300	C₄	C₇×#550	1	1	yes	characteristic
301	C₄	C₇×#559	1	1	yes	characteristic
302	C₂	C₇×#2366	1	1	yes	characteristic
303	C₂	C₂×C₇×#441	1	1	yes	characteristic
304	C₂	C₇×#2950	1	1	yes	characteristic
305	C₂	C₇×#3276	1	2	yes	—
306	C₂	C₇×#3398	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
13. C₇×#494	C₄	semidirect
30. #23187	C₇	direct (N × H)
116. #494	C₄×C₇	semidirect
208. C₄×C₇	#441	semidirect
286. C₇	#23187	direct (N × H)
297. C₄	C₇×#441	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#494	2
C₂×C₇×#441	2
C₇×#2365	2
C₇×#1981	2
C₇×#1971	2
C₇×#1878	2
C₄×C₇×#106	2
C₇×#1770	2
#23187	7

Subgroups — C7×#23187 #13200403

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#23187 #13200403