Subgroups — C₇×#2452 #11741761

Order 896.

620 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic
4	C₇×#396	C₂	1	2	yes	maximal, complemented
5	C₇×#393	C₂	1	2	yes	maximal, complemented
6	C₇×#390	C₂	1	1	yes	maximal, characteristic
7	C₇×#390	C₂	1	1	yes	maximal, characteristic
8	C₇×#389	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#389	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#385	C₂	1	2	yes	maximal, complemented
11	C₄×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₇×Q₈	C₂²	1	2	yes	—
13	C₂²×C₇×D₄	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₇	C₂²	1	2	yes	—
15	C₇×C₄⋊Q₈	—	2	4	no	—
16	C₇×C₂²⋊Q₈	—	2	8	no	—
17	C₄×C₇×D₄	—	2	4	no	—
18	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
21	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
22	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
26	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
27	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
28	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
29	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
30	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
31	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
32	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
33	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
34	C₇×#95	C₂²	1	1	yes	characteristic
35	C₇×#95	C₂²	1	1	yes	characteristic
36	C₇×#95	C₂²	1	2	yes	—
37	C₇×#95	C₂²	1	2	yes	—
38	#2452	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	C₂⁴×C₇	C₂³	1	2	yes	—
40	C₂×C₇×Q₈	—	2	4	no	—
41	C₂×C₇×Q₈	—	2	4	no	—
42	C₂×C₇×Q₈	—	4	2	no	—
43	C₂×C₇×D₄	—	2	2	no	—
44	C₂×C₇×D₄	—	2	2	no	—
45	C₂×C₇×D₄	Q₈	1	4	yes	complemented
46	C₂²×C₄×C₇	—	2	4	no	—
47	C₂²×C₄×C₇	—	2	4	no	—
48	C₂²×C₄×C₇	—	4	2	no	—
49	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₇	C₂³	1	2	yes	—
52	C₂²×C₄×C₇	C₂³	1	2	yes	—
53	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₇	C₂³	1	2	yes	—
56	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
59	C₇×C₄⋊C₄	—	2	4	no	—
60	C₇×C₄⋊C₄	—	2	4	no	—
61	C₇×C₄⋊C₄	—	2	4	no	—
62	C₇×C₄⋊C₄	—	2	2	no	—
63	C₇×C₄⋊C₄	—	2	2	no	—
64	C₇×C₄⋊C₄	—	2	2	no	—
65	C₇×C₄⋊C₄	D₄	1	4	yes	—
66	C₇×C₄⋊C₄	D₄	1	4	yes	complemented
67	C₇×C₂²⋊C₄	—	2	4	no	—
68	C₇×C₂²⋊C₄	—	2	4	no	—
69	C₇×C₂²⋊C₄	—	2	4	no	—
70	C₇×C₂²⋊C₄	—	2	4	no	—
71	C₇×C₂²⋊C₄	—	4	2	no	—
72	C₄²×C₇	—	2	2	no	—
73	C₄²×C₇	—	2	2	no	—
74	C₄²×C₇	—	2	2	no	—
75	C₂×C₄⋊Q₈	C₂×C₇	1	1	yes	characteristic, complemented
76	C₂×C₂²⋊Q₈	C₂×C₇	1	2	yes	complemented
77	C₂×C₄×D₄	C₂×C₇	1	1	yes	characteristic
78	#396	C₂×C₇	1	2	yes	complemented
79	#393	C₂×C₇	1	2	yes	complemented
80	#390	C₂×C₇	1	1	yes	characteristic
81	#390	C₂×C₇	1	1	yes	characteristic
82	#389	C₂×C₇	1	1	yes	characteristic, complemented
83	#389	C₂×C₇	1	1	yes	characteristic, complemented
84	#385	C₂×C₇	1	2	yes	complemented
85	C₄×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
86	C₂³×C₇	—	2	2	no	—
87	C₂³×C₇	—	2	2	no	—
88	C₂³×C₇	—	2	2	no	—
89	C₂³×C₇	—	2	2	no	—
90	C₂³×C₇	—	2	2	no	—
91	C₂³×C₇	—	2	2	no	—
92	C₂³×C₇	C₂×Q₈	1	4	yes	—
93	C₂³×C₇	C₂⁴	1	1	yes	characteristic
94	C₇×Q₈	—	4	8	no	—
95	C₇×D₄	—	2	8	no	—
96	C₂×C₄×C₇	—	2	1	no	—
97	C₂×C₄×C₇	—	2	1	no	—
98	C₂×C₄×C₇	—	2	1	no	—
99	C₂×C₄×C₇	—	2	2	no	—
100	C₂×C₄×C₇	—	2	1	no	—
101	C₂×C₄×C₇	—	2	1	no	—
102	C₂×C₄×C₇	—	2	1	no	—
103	C₂×C₄×C₇	—	2	2	no	—
104	C₂×C₄×C₇	—	2	1	no	—
105	C₂×C₄×C₇	—	2	1	no	—
106	C₂×C₄×C₇	—	2	1	no	—
107	C₂×C₄×C₇	—	2	2	no	—
108	C₂×C₄×C₇	—	2	2	no	—
109	C₂×C₄×C₇	—	2	2	no	—
110	C₂×C₄×C₇	—	2	1	no	—
111	C₂×C₄×C₇	—	2	2	no	—
112	C₂×C₄×C₇	—	2	2	no	—
113	C₂×C₄×C₇	—	2	1	no	—
114	C₂×C₄×C₇	—	2	1	no	—
115	C₂×C₄×C₇	—	2	2	no	—
116	C₂×C₄×C₇	—	2	1	no	—
117	C₂×C₄×C₇	—	2	1	no	—
118	C₂×C₄×C₇	—	4	8	no	—
119	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
120	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
121	C₂×C₄×C₇	C₂×D₄	1	4	yes	—
122	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
123	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
124	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
125	C₂×C₄×C₇	#41	1	2	yes	—
126	C₂×C₄×C₇	#41	1	2	yes	—
127	C₂²×Q₈	C₂²×C₇	1	2	yes	—
128	C₂²×D₄	C₂²×C₇	1	1	yes	characteristic
129	C₂³×C₄	C₂²×C₇	1	2	yes	—
130	C₄⋊Q₈	—	2	4	no	—
131	C₂²⋊Q₈	—	2	8	no	—
132	C₄×D₄	—	2	4	no	—
133	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
134	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
135	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
136	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
137	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
138	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
139	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
140	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
141	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
142	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
143	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
144	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
145	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
146	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
147	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
148	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
149	#95	C₂²×C₇	1	1	yes	characteristic
150	#95	C₂²×C₇	1	1	yes	characteristic
151	#95	C₂²×C₇	1	2	yes	—
152	#95	C₂²×C₇	1	2	yes	—
153	C₂²×C₇	—	2	4	no	—
154	C₂²×C₇	—	2	4	no	—
155	C₂²×C₇	—	2	4	no	—
156	C₂²×C₇	—	4	2	no	—
157	C₂²×C₇	—	4	2	no	—
158	C₂²×C₇	—	4	2	no	—
159	C₂²×C₇	—	4	2	no	—
160	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
161	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
162	C₂²×C₇	C₂²×Q₈	1	1	yes	characteristic
163	C₂²×C₇	C₂×#41	1	1	yes	characteristic
164	C₂²×C₇	C₂×#41	1	1	yes	characteristic
165	C₂²×C₇	H(5,𝔽₂)	1	1	yes	characteristic
166	C₂²×C₇	2₋¹⁺⁴	1	1	yes	characteristic
167	C₄×C₇	—	2	2	no	—
168	C₄×C₇	—	2	2	no	—
169	C₄×C₇	—	2	4	no	—
170	C₄×C₇	—	2	2	no	—
171	C₄×C₇	—	2	2	no	—
172	C₄×C₇	—	4	2	no	—
173	C₄×C₇	—	4	1	no	—
174	C₄×C₇	—	4	2	no	—
175	C₄×C₇	—	4	1	no	—
176	C₄×C₇	C₂²⋊Q₈	1	4	yes	complemented
177	C₂⁴	C₂³×C₇	1	2	yes	—
178	C₂×Q₈	—	2	4	no	—
179	C₂×Q₈	—	2	4	no	—
180	C₂×Q₈	—	4	2	no	—
181	C₂×D₄	—	2	2	no	—
182	C₂×D₄	—	2	2	no	—
183	C₂×D₄	C₇×Q₈	1	4	yes	complemented
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	—	4	2	no	—
187	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
188	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
189	C₂²×C₄	C₂³×C₇	1	2	yes	—
190	C₂²×C₄	C₂³×C₇	1	2	yes	—
191	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
192	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
193	C₂²×C₄	C₂³×C₇	1	2	yes	—
194	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
195	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
196	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	—	2	4	no	—
199	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	C₇×D₄	1	4	yes	—
204	C₄⋊C₄	C₇×D₄	1	4	yes	complemented
205	C₂²⋊C₄	—	2	4	no	—
206	C₂²⋊C₄	—	2	4	no	—
207	C₂²⋊C₄	—	2	4	no	—
208	C₂²⋊C₄	—	2	4	no	—
209	C₂²⋊C₄	—	4	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₂×C₇	—	4	4	no	—
214	C₂×C₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
215	C₂×C₇	#532	1	1	yes	characteristic
216	C₂×C₇	#542	1	1	yes	characteristic
217	C₂×C₇	D₄²	1	1	yes	characteristic
218	C₂×C₇	#548	1	1	yes	characteristic
219	C₂×C₇	D₄×Q₈	1	1	yes	characteristic
220	C₂×C₇	#554	1	1	yes	characteristic
221	C₂³	—	2	2	no	—
222	C₂³	—	2	2	no	—
223	C₂³	—	2	2	no	—
224	C₂³	—	2	2	no	—
225	C₂³	—	2	2	no	—
226	C₂³	—	2	2	no	—
227	C₂³	C₂×C₇×Q₈	1	4	yes	—
228	C₂³	C₂⁴×C₇	1	1	yes	characteristic
229	Q₈	—	4	8	no	—
230	D₄	—	2	8	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	4	8	no	—
254	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
255	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
256	C₂×C₄	C₂×C₇×D₄	1	4	yes	—
257	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
258	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
259	C₂×C₄	C₂×C₇×Q₈	1	2	yes	—
260	C₂×C₄	C₇×#41	1	2	yes	—
261	C₂×C₄	C₇×#41	1	2	yes	—
262	C₇	#2452	1	1	yes	characteristic, direct product factor, Sylow
263	C₂²	—	2	4	no	—
264	C₂²	—	2	4	no	—
265	C₂²	—	2	4	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	2	no	—
270	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
271	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
272	C₂²	C₂²×C₇×Q₈	1	1	yes	characteristic
273	C₂²	C₂×C₇×#41	1	1	yes	characteristic
274	C₂²	C₂×C₇×#41	1	1	yes	characteristic
275	C₂²	C₇×H(5,𝔽₂)	1	1	yes	characteristic
276	C₂²	C₇×2₋¹⁺⁴	1	1	yes	characteristic
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	—	2	4	no	—
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	4	2	no	—
283	C₄	—	4	1	no	—
284	C₄	—	4	2	no	—
285	C₄	—	4	1	no	—
286	C₄	C₇×C₂²⋊Q₈	1	4	yes	complemented
287	C₂	—	4	4	no	—
288	C₂	C₂×C₇×C₂²⋊Q₈	1	1	yes	characteristic
289	C₂	C₇×#532	1	1	yes	characteristic
290	C₂	C₇×#542	1	1	yes	characteristic
291	C₂	C₇×D₄²	1	1	yes	characteristic
292	C₂	C₇×#548	1	1	yes	characteristic
293	C₂	C₇×D₄×Q₈	1	1	yes	characteristic
294	C₂	C₇×#554	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×C₄⋊Q₈	C₂	semidirect
2. C₂×C₇×C₂²⋊Q₈	C₂	semidirect
4. C₇×#396	C₂	semidirect
5. C₇×#393	C₂	semidirect
8. C₇×#389	C₂	semidirect
9. C₇×#389	C₂	semidirect
10. C₇×#385	C₂	semidirect
11. C₄×C₇×C₄⋊C₄	C₂	semidirect
38. #2452	C₇	direct (N × H)
45. C₂×C₇×D₄	Q₈	semidirect
66. C₇×C₄⋊C₄	D₄	semidirect
75. C₂×C₄⋊Q₈	C₂×C₇	semidirect
76. C₂×C₂²⋊Q₈	C₂×C₇	semidirect
78. #396	C₂×C₇	semidirect
79. #393	C₂×C₇	semidirect
82. #389	C₂×C₇	semidirect
83. #389	C₂×C₇	semidirect
84. #385	C₂×C₇	semidirect
85. C₄×C₄⋊C₄	C₂×C₇	semidirect
176. C₄×C₇	C₂²⋊Q₈	semidirect
183. C₂×D₄	C₇×Q₈	semidirect
204. C₄⋊C₄	C₇×D₄	semidirect
262. C₇	#2452	direct (N × H)
286. C₄	C₇×C₂²⋊Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×C₄⋊Q₈	2
C₂×C₇×C₂²⋊Q₈	2
C₂×C₄×C₇×D₄	2
C₇×#396	2
C₇×#393	2
C₇×#390	2
C₇×#389	2
C₇×#385	2
C₄×C₇×C₄⋊C₄	2
#2452	7

Subgroups — C7×#2452 #11741761

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2452 #11741761