Subgroups — C₇×#11258 #13188474

Order 1792.

654 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	C₂²×C₇×#108	C₂	1	1	yes	maximal, characteristic
2	C₇×#1735	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×#339	C₂	1	2	yes	maximal
4	C₇×#1729	C₂	1	2	yes	maximal
5	C₂×C₇×#336	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
7	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₇×#117	C₂²	1	2	yes	—
10	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
11	C₂×C₇×#108	—	2	4	no	—
12	C₂×C₇×#108	—	2	2	no	—
13	C₇×#340	C₄	1	4	yes	complemented
14	C₇×#339	—	2	4	no	—
15	C₇×#336	—	2	2	no	—
16	C₇×#336	—	2	4	no	—
17	#11258	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂³×C₄×C₇	—	2	1	no	—
19	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
20	C₂×C₇×M₁₆	—	2	2	no	—
21	C₂×C₇×M₁₆	—	2	2	no	—
22	C₂×C₇×M₁₆	—	2	2	no	—
23	C₂×C₇×M₁₆	—	2	2	no	—
24	C₂×C₇×M₁₆	—	2	2	no	—
25	C₂×C₇×M₁₆	D₄	1	4	yes	—
26	C₂²×C₇×C₈	—	2	2	no	—
27	C₂²×C₇×C₈	—	2	2	no	—
28	C₂²×C₇×C₈	—	2	1	no	—
29	C₂²×C₇×C₈	—	2	2	no	—
30	C₂²×C₇×C₈	—	2	1	no	—
31	C₂²×C₇×C₈	—	2	2	no	—
32	C₂²×C₇×C₈	—	2	2	no	—
33	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
34	C₂²×C₇×C₈	D₄	1	2	yes	—
35	C₂²×C₇×C₈	D₄	1	2	yes	—
36	C₂²×C₇×C₈	Q₈	1	2	yes	—
37	C₇×#117	—	2	2	no	—
38	C₇×#117	—	2	2	no	—
39	C₇×#117	—	2	2	no	—
40	C₇×#117	—	2	2	no	—
41	C₂×C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
42	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
43	C₂×C₄²×C₇	—	2	2	no	—
44	C₂×C₄²×C₇	—	2	2	no	—
45	C₂×C₄²×C₇	—	2	1	no	—
46	C₂×C₄²×C₇	—	2	2	no	—
47	C₂×C₄²×C₇	Q₈	1	2	yes	—
48	C₇×#108	—	2	8	no	—
49	C₂²×#108	C₂×C₇	1	1	yes	characteristic
50	#1735	C₂×C₇	1	1	yes	characteristic
51	C₂×#339	C₂×C₇	1	2	yes	—
52	#1729	C₂×C₇	1	2	yes	—
53	C₂×#336	C₂×C₇	1	1	yes	characteristic
54	C₂⁴×C₇	#41	1	1	yes	characteristic
55	C₂²×C₄×C₇	—	2	4	no	—
56	C₂²×C₄×C₇	—	2	2	no	—
57	C₂²×C₄×C₇	—	2	4	no	—
58	C₂²×C₄×C₇	—	4	2	no	—
59	C₂²×C₄×C₇	—	4	2	no	—
60	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₇	#41	1	2	yes	—
67	C₂²×C₄×C₇	#41	1	2	yes	—
68	C₂²×C₄×C₇	#41	1	2	yes	—
69	C₂²×C₄×C₇	#41	1	1	yes	characteristic
70	C₇×M₁₆	—	2	4	no	—
71	C₇×M₁₆	—	2	4	no	—
72	C₇×M₁₆	—	4	2	no	—
73	C₇×M₁₆	—	4	2	no	—
74	C₂×C₇×C₈	—	2	4	no	—
75	C₂×C₇×C₈	—	2	2	no	—
76	C₂×C₇×C₈	—	2	4	no	—
77	C₂×C₇×C₈	—	2	2	no	—
78	C₂×C₇×C₈	—	2	2	no	—
79	C₂×C₇×C₈	—	2	4	no	—
80	C₂×C₇×C₈	—	2	4	no	—
81	C₂×C₇×C₈	—	4	2	no	—
82	C₂×C₇×C₈	—	4	2	no	—
83	C₂×C₇×C₈	—	4	1	no	—
84	C₂×C₇×C₈	—	4	1	no	—
85	C₂×C₇×C₈	—	4	2	no	—
86	C₂×C₇×C₈	C₄⋊C₄	1	4	yes	—
87	C₇×C₄⋊C₄	—	2	4	no	—
88	C₇×C₄⋊C₄	—	2	4	no	—
89	C₇×C₂²⋊C₄	—	4	2	no	—
90	C₇×C₂²⋊C₄	—	4	2	no	—
91	C₄²×C₇	—	2	4	no	—
92	C₄²×C₇	—	4	1	no	—
93	C₄²×C₇	—	4	2	no	—
94	C₄²×C₇	—	4	2	no	—
95	C₄²×C₇	—	4	1	no	—
96	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
97	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
98	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
99	C₂×#117	C₂²×C₇	1	2	yes	—
100	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
101	C₂×#108	—	2	4	no	—
102	C₂×#108	—	2	2	no	—
103	#340	C₄×C₇	1	4	yes	complemented
104	#339	—	2	4	no	—
105	#336	—	2	2	no	—
106	#336	—	2	4	no	—
107	C₂³×C₇	—	2	2	no	—
108	C₂³×C₇	—	2	2	no	—
109	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
110	C₂³×C₇	C₂²⋊Q₈	1	1	yes	characteristic
111	C₂³×C₇	#124	1	2	yes	—
112	C₂³×C₇	#125	1	1	yes	characteristic
113	C₂³×C₇	#126	1	2	yes	—
114	C₂×C₄×C₇	—	2	4	no	—
115	C₂×C₄×C₇	—	4	8	no	—
116	C₂×C₄×C₇	—	4	2	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	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₇	#117	1	2	yes	—
123	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₇	C₄×Q₈	1	1	yes	characteristic
125	C₂×C₄×C₇	#120	1	1	yes	characteristic
126	C₂×C₄×C₇	#121	1	1	yes	characteristic
127	C₂×C₄×C₇	#121	1	2	yes	—
128	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
129	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
130	C₂×C₄×C₇	#123	1	1	yes	characteristic
131	C₂×C₄×C₇	#123	1	2	yes	—
132	C₂×C₄×C₇	#123	1	2	yes	—
133	C₂×C₄×C₇	#124	1	1	yes	characteristic
134	C₂×C₄×C₇	#124	1	1	yes	characteristic
135	C₂×C₄×C₇	#125	1	2	yes	—
136	C₂×C₄×C₇	#125	1	2	yes	—
137	C₂×C₄×C₇	#126	1	2	yes	—
138	C₂×C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
139	C₇×C₈	—	2	4	no	—
140	C₇×C₈	—	4	4	no	—
141	C₇×C₈	—	4	2	no	—
142	C₂³×C₄	—	2	1	no	—
143	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
144	C₂×M₁₆	—	2	2	no	—
145	C₂×M₁₆	—	2	2	no	—
146	C₂×M₁₆	—	2	2	no	—
147	C₂×M₁₆	—	2	2	no	—
148	C₂×M₁₆	—	2	2	no	—
149	C₂×M₁₆	C₇×D₄	1	4	yes	—
150	C₂²×C₈	—	2	2	no	—
151	C₂²×C₈	—	2	2	no	—
152	C₂²×C₈	—	2	1	no	—
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	—	2	1	no	—
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
158	C₂²×C₈	C₇×D₄	1	2	yes	—
159	C₂²×C₈	C₇×D₄	1	2	yes	—
160	C₂²×C₈	C₇×Q₈	1	2	yes	—
161	#117	—	2	2	no	—
162	#117	—	2	2	no	—
163	#117	—	2	2	no	—
164	#117	—	2	2	no	—
165	C₂×C₄⋊C₄	C₂×C₄×C₇	1	4	yes	—
166	C₂×C₂²⋊C₄	—	2	2	no	—
167	C₂×C₄²	—	2	2	no	—
168	C₂×C₄²	—	2	2	no	—
169	C₂×C₄²	—	2	2	no	—
170	C₂×C₄²	—	2	1	no	—
171	C₂×C₄²	C₇×Q₈	1	2	yes	—
172	#108	—	2	8	no	—
173	C₂²×C₇	—	2	4	no	—
174	C₂²×C₇	—	2	4	no	—
175	C₂²×C₇	—	2	1	no	—
176	C₂²×C₇	—	2	1	no	—
177	C₂²×C₇	—	2	1	no	—
178	C₂²×C₇	—	2	1	no	—
179	C₂²×C₇	#388	1	2	yes	—
180	C₂²×C₇	#389	1	1	yes	characteristic
181	C₂²×C₇	#393	1	1	yes	characteristic
182	C₂²×C₇	#400	1	2	yes	—
183	C₂²×C₇	#400	1	1	yes	characteristic
184	C₂²×C₇	#443	1	2	yes	—
185	C₂²×C₇	#444	1	2	yes	—
186	C₄×C₇	—	4	4	no	—
187	C₄×C₇	—	8	4	no	—
188	C₄×C₇	#382	1	1	yes	characteristic
189	C₄×C₇	#385	1	1	yes	characteristic
190	C₄×C₇	#387	1	2	yes	—
191	C₄×C₇	#394	1	1	yes	characteristic
192	C₄×C₇	#397	1	1	yes	characteristic
193	C₄×C₇	#398	1	2	yes	—
194	C₂⁴	C₇×#41	1	1	yes	characteristic
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	4	2	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
203	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
206	C₂²×C₄	C₇×#41	1	2	yes	—
207	C₂²×C₄	C₇×#41	1	2	yes	—
208	C₂²×C₄	C₇×#41	1	2	yes	—
209	C₂²×C₄	C₇×#41	1	1	yes	characteristic
210	M₁₆	—	2	4	no	—
211	M₁₆	—	2	4	no	—
212	M₁₆	—	4	2	no	—
213	M₁₆	—	4	2	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	C₇×C₄⋊C₄	1	4	yes	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	2	4	no	—
229	C₂²⋊C₄	—	4	2	no	—
230	C₂²⋊C₄	—	4	2	no	—
231	C₄²	—	2	4	no	—
232	C₄²	—	4	1	no	—
233	C₄²	—	4	2	no	—
234	C₄²	—	4	2	no	—
235	C₄²	—	4	1	no	—
236	C₂×C₇	—	2	1	no	—
237	C₂×C₇	—	2	2	no	—
238	C₂×C₇	—	2	1	no	—
239	C₂×C₇	#1442	1	1	yes	characteristic
240	C₂×C₇	#1929	1	4	yes	—
241	C₂×C₇	#1945	1	1	yes	characteristic
242	C₂×C₇	#1946	1	1	yes	characteristic
243	C₂³	—	2	2	no	—
244	C₂³	—	2	2	no	—
245	C₂³	C₄×C₇×D₄	1	1	yes	characteristic
246	C₂³	C₇×C₂²⋊Q₈	1	1	yes	characteristic
247	C₂³	C₇×#124	1	2	yes	—
248	C₂³	C₇×#125	1	1	yes	characteristic
249	C₂³	C₇×#126	1	2	yes	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	4	8	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
258	C₂×C₄	C₇×#117	1	2	yes	—
259	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
260	C₂×C₄	C₄×C₇×Q₈	1	1	yes	characteristic
261	C₂×C₄	C₇×#120	1	1	yes	characteristic
262	C₂×C₄	C₇×#121	1	1	yes	characteristic
263	C₂×C₄	C₇×#121	1	2	yes	—
264	C₂×C₄	C₇×C₂²⋊Q₈	1	2	yes	—
265	C₂×C₄	C₇×C₂²⋊Q₈	1	2	yes	—
266	C₂×C₄	C₇×#123	1	1	yes	characteristic
267	C₂×C₄	C₇×#123	1	2	yes	—
268	C₂×C₄	C₇×#123	1	2	yes	—
269	C₂×C₄	C₇×#124	1	1	yes	characteristic
270	C₂×C₄	C₇×#124	1	1	yes	characteristic
271	C₂×C₄	C₇×#125	1	2	yes	—
272	C₂×C₄	C₇×#125	1	2	yes	—
273	C₂×C₄	C₇×#126	1	2	yes	—
274	C₂×C₄	C₇×C₄⋊Q₈	1	1	yes	characteristic
275	C₈	—	2	4	no	—
276	C₈	—	4	4	no	—
277	C₈	—	4	2	no	—
278	C₇	#11258	1	1	yes	characteristic, direct product factor, Sylow
279	C₂²	—	2	4	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	C₇×#388	1	2	yes	—
286	C₂²	C₇×#389	1	1	yes	characteristic
287	C₂²	C₇×#393	1	1	yes	characteristic
288	C₂²	C₇×#400	1	2	yes	—
289	C₂²	C₇×#400	1	1	yes	characteristic
290	C₂²	C₇×#443	1	2	yes	—
291	C₂²	C₇×#444	1	2	yes	—
292	C₄	—	4	4	no	—
293	C₄	—	8	4	no	—
294	C₄	C₇×#382	1	1	yes	characteristic
295	C₄	C₇×#385	1	1	yes	characteristic
296	C₄	C₇×#387	1	2	yes	—
297	C₄	C₇×#394	1	1	yes	characteristic
298	C₄	C₇×#397	1	1	yes	characteristic
299	C₄	C₇×#398	1	2	yes	—
300	C₂	—	2	1	no	—
301	C₂	—	2	2	no	—
302	C₂	—	2	1	no	—
303	C₂	C₇×#1442	1	1	yes	characteristic
304	C₂	C₇×#1929	1	4	yes	—
305	C₂	C₇×#1945	1	1	yes	characteristic
306	C₂	C₇×#1946	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₇	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂³×C₄	C₂³×C₇
socle	C₂³×C₇	C₄×D₄
prime radical	C₂⁴×C₇	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ C₂³×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₇×#340	C₄	semidirect
17. #11258	C₇	direct (N × H)
103. #340	C₄×C₇	semidirect
278. C₇	#11258	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×#108	2
C₇×#1735	2
C₂×C₇×#339	2
C₇×#1729	2
C₂×C₇×#336	2
#11258	7

Subgroups — C7×#11258 #13188474

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#11258 #13188474