Subgroups — C₇×#2538 #11741847

Order 896.

620 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic
4	C₇×#396	C₂	1	4	yes	maximal, complemented
5	C₇×#390	C₂	1	2	yes	maximal
6	C₇×#389	C₂	1	2	yes	maximal, complemented
7	C₇×#384	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#380	C₂	1	2	yes	maximal, complemented
9	C₂²×C₇×Q₈	C₂²	1	1	yes	characteristic
10	C₂²×C₇×D₄	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₇	C₂²	1	2	yes	—
12	C₇×C₂²⋊Q₈	—	2	8	no	—
13	C₄×C₇×Q₈	—	2	4	no	—
14	C₄×C₇×D₄	—	2	4	no	—
15	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₇×C₄⋊C₄	C₂²	1	4	yes	—
21	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
22	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₂×C₇×C₂²⋊C₄	C₂²	1	4	yes	—
24	C₂×C₇×C₂²⋊C₄	C₂²	1	4	yes	—
25	C₂×C₄²×C₇	C₂²	1	2	yes	—
26	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
27	C₇×#95	C₂²	1	4	yes	—
28	#2538	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₂⁴×C₇	C₂³	1	2	yes	—
30	C₂×C₇×Q₈	—	2	2	no	—
31	C₂×C₇×Q₈	—	2	2	no	—
32	C₂×C₇×Q₈	D₄	1	4	yes	complemented
33	C₂×C₇×D₄	—	2	2	no	—
34	C₂×C₇×D₄	—	2	2	no	—
35	C₂×C₇×D₄	Q₈	1	4	yes	complemented
36	C₂²×C₄×C₇	—	2	4	no	—
37	C₂²×C₄×C₇	—	2	4	no	—
38	C₂²×C₄×C₇	—	2	4	no	—
39	C₂²×C₄×C₇	C₂³	1	2	yes	—
40	C₂²×C₄×C₇	C₂³	1	2	yes	—
41	C₂²×C₄×C₇	C₂³	1	4	yes	—
42	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₇	C₂³	1	2	yes	—
44	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
46	C₇×C₄⋊C₄	—	2	2	no	—
47	C₇×C₄⋊C₄	—	2	4	no	—
48	C₇×C₄⋊C₄	—	2	2	no	—
49	C₇×C₄⋊C₄	—	2	4	no	—
50	C₇×C₄⋊C₄	—	2	8	no	—
51	C₇×C₄⋊C₄	—	2	4	no	—
52	C₇×C₄⋊C₄	—	4	2	no	—
53	C₇×C₂²⋊C₄	—	2	4	no	—
54	C₇×C₂²⋊C₄	—	2	8	no	—
55	C₇×C₂²⋊C₄	—	4	4	no	—
56	C₄²×C₇	—	2	4	no	—
57	C₄²×C₇	D₄	1	4	yes	—
58	C₂×C₂²⋊Q₈	C₂×C₇	1	2	yes	complemented
59	C₂×C₄×Q₈	C₂×C₇	1	1	yes	characteristic, complemented
60	C₂×C₄×D₄	C₂×C₇	1	1	yes	characteristic
61	#396	C₂×C₇	1	4	yes	complemented
62	#390	C₂×C₇	1	2	yes	—
63	#389	C₂×C₇	1	2	yes	complemented
64	#384	C₂×C₇	1	1	yes	characteristic, complemented
65	#380	C₂×C₇	1	2	yes	complemented
66	C₂³×C₇	—	2	2	no	—
67	C₂³×C₇	—	2	2	no	—
68	C₂³×C₇	—	2	2	no	—
69	C₂³×C₇	—	2	2	no	—
70	C₂³×C₇	—	2	2	no	—
71	C₂³×C₇	—	2	2	no	—
72	C₂³×C₇	C₂×Q₈	1	4	yes	—
73	C₂³×C₇	C₂⁴	1	1	yes	characteristic
74	C₇×Q₈	—	2	8	no	—
75	C₇×D₄	—	2	8	no	—
76	C₂×C₄×C₇	—	2	2	no	—
77	C₂×C₄×C₇	—	2	2	no	—
78	C₂×C₄×C₇	—	2	1	no	—
79	C₂×C₄×C₇	—	2	1	no	—
80	C₂×C₄×C₇	—	2	2	no	—
81	C₂×C₄×C₇	—	2	4	no	—
82	C₂×C₄×C₇	—	2	2	no	—
83	C₂×C₄×C₇	—	2	2	no	—
84	C₂×C₄×C₇	—	2	4	no	—
85	C₂×C₄×C₇	—	2	2	no	—
86	C₂×C₄×C₇	—	2	2	no	—
87	C₂×C₄×C₇	—	2	4	no	—
88	C₂×C₄×C₇	—	2	2	no	—
89	C₂×C₄×C₇	—	4	8	no	—
90	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
91	C₂×C₄×C₇	C₂×D₄	1	4	yes	—
92	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
93	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
96	C₂×C₄×C₇	#41	1	2	yes	—
97	C₂×C₄×C₇	#41	1	2	yes	—
98	C₂²×Q₈	C₂²×C₇	1	1	yes	characteristic
99	C₂²×D₄	C₂²×C₇	1	1	yes	characteristic
100	C₂³×C₄	C₂²×C₇	1	2	yes	—
101	C₂²⋊Q₈	—	2	8	no	—
102	C₄×Q₈	—	2	4	no	—
103	C₄×D₄	—	2	4	no	—
104	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
105	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
106	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
107	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
108	C₂×C₄⋊C₄	C₂²×C₇	1	4	yes	—
109	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
110	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
111	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
112	C₂×C₂²⋊C₄	C₂²×C₇	1	4	yes	—
113	C₂×C₂²⋊C₄	C₂²×C₇	1	4	yes	—
114	C₂×C₄²	C₂²×C₇	1	2	yes	—
115	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
116	#95	C₂²×C₇	1	4	yes	—
117	C₂²×C₇	—	2	4	no	—
118	C₂²×C₇	—	2	4	no	—
119	C₂²×C₇	—	2	4	no	—
120	C₂²×C₇	—	4	2	no	—
121	C₂²×C₇	—	4	2	no	—
122	C₂²×C₇	—	4	2	no	—
123	C₂²×C₇	—	4	2	no	—
124	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
125	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
126	C₂²×C₇	C₂²×Q₈	1	1	yes	characteristic
127	C₂²×C₇	C₂×#41	1	1	yes	characteristic
128	C₂²×C₇	C₂×#41	1	1	yes	characteristic
129	C₂²×C₇	H(5,𝔽₂)	1	1	yes	characteristic
130	C₂²×C₇	2₋¹⁺⁴	1	1	yes	characteristic
131	C₄×C₇	—	2	4	no	—
132	C₄×C₇	—	2	2	no	—
133	C₄×C₇	—	4	2	no	—
134	C₄×C₇	—	4	2	no	—
135	C₄×C₇	—	4	4	no	—
136	C₄×C₇	#121	1	4	yes	—
137	C₄×C₇	C₂²⋊Q₈	1	4	yes	complemented
138	C₂⁴	C₂³×C₇	1	2	yes	—
139	C₂×Q₈	—	2	2	no	—
140	C₂×Q₈	—	2	2	no	—
141	C₂×Q₈	C₇×D₄	1	4	yes	complemented
142	C₂×D₄	—	2	2	no	—
143	C₂×D₄	—	2	2	no	—
144	C₂×D₄	C₇×Q₈	1	4	yes	complemented
145	C₂²×C₄	—	2	4	no	—
146	C₂²×C₄	—	2	4	no	—
147	C₂²×C₄	—	2	4	no	—
148	C₂²×C₄	C₂³×C₇	1	2	yes	—
149	C₂²×C₄	C₂³×C₇	1	2	yes	—
150	C₂²×C₄	C₂³×C₇	1	4	yes	—
151	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
152	C₂²×C₄	C₂³×C₇	1	2	yes	—
153	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
154	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	2	4	no	—
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	—	2	4	no	—
159	C₄⋊C₄	—	2	8	no	—
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₂²⋊C₄	—	2	4	no	—
163	C₂²⋊C₄	—	2	8	no	—
164	C₂²⋊C₄	—	4	4	no	—
165	C₄²	—	2	4	no	—
166	C₄²	C₇×D₄	1	4	yes	—
167	C₂×C₇	—	4	4	no	—
168	C₂×C₇	C₂×#121	1	1	yes	characteristic
169	C₂×C₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
170	C₂×C₇	#537	1	1	yes	characteristic
171	C₂×C₇	D₄×Q₈	1	1	yes	characteristic
172	C₂×C₇	#550	1	1	yes	characteristic
173	C₂×C₇	#554	1	1	yes	characteristic
174	C₂×C₇	#556	1	1	yes	characteristic
175	C₂³	—	2	2	no	—
176	C₂³	—	2	2	no	—
177	C₂³	—	2	2	no	—
178	C₂³	—	2	2	no	—
179	C₂³	—	2	2	no	—
180	C₂³	—	2	2	no	—
181	C₂³	C₂×C₇×Q₈	1	4	yes	—
182	C₂³	C₂⁴×C₇	1	1	yes	characteristic
183	Q₈	—	2	8	no	—
184	D₄	—	2	8	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	4	8	no	—
199	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
200	C₂×C₄	C₂×C₇×D₄	1	4	yes	—
201	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
202	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
203	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
204	C₂×C₄	C₂×C₇×Q₈	1	2	yes	—
205	C₂×C₄	C₇×#41	1	2	yes	—
206	C₂×C₄	C₇×#41	1	2	yes	—
207	C₇	#2538	1	1	yes	characteristic, direct product factor, Sylow
208	C₂²	—	2	4	no	—
209	C₂²	—	2	4	no	—
210	C₂²	—	2	4	no	—
211	C₂²	—	4	2	no	—
212	C₂²	—	4	2	no	—
213	C₂²	—	4	2	no	—
214	C₂²	—	4	2	no	—
215	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
216	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
217	C₂²	C₂²×C₇×Q₈	1	1	yes	characteristic
218	C₂²	C₂×C₇×#41	1	1	yes	characteristic
219	C₂²	C₂×C₇×#41	1	1	yes	characteristic
220	C₂²	C₇×H(5,𝔽₂)	1	1	yes	characteristic
221	C₂²	C₇×2₋¹⁺⁴	1	1	yes	characteristic
222	C₄	—	2	4	no	—
223	C₄	—	2	2	no	—
224	C₄	—	4	2	no	—
225	C₄	—	4	2	no	—
226	C₄	—	4	4	no	—
227	C₄	C₇×#121	1	4	yes	—
228	C₄	C₇×C₂²⋊Q₈	1	4	yes	complemented
229	C₂	—	4	4	no	—
230	C₂	C₂×C₇×#121	1	1	yes	characteristic
231	C₂	C₂×C₇×C₂²⋊Q₈	1	1	yes	characteristic
232	C₂	C₇×#537	1	1	yes	characteristic
233	C₂	C₇×D₄×Q₈	1	1	yes	characteristic
234	C₂	C₇×#550	1	1	yes	characteristic
235	C₂	C₇×#554	1	1	yes	characteristic
236	C₂	C₇×#556	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×C₂²⋊Q₈	C₂	semidirect
2. C₂×C₄×C₇×Q₈	C₂	semidirect
4. C₇×#396	C₂	semidirect
6. C₇×#389	C₂	semidirect
7. C₇×#384	C₂	semidirect
8. C₇×#380	C₂	semidirect
28. #2538	C₇	direct (N × H)
32. C₂×C₇×Q₈	D₄	semidirect
35. C₂×C₇×D₄	Q₈	semidirect
58. C₂×C₂²⋊Q₈	C₂×C₇	semidirect
59. C₂×C₄×Q₈	C₂×C₇	semidirect
61. #396	C₂×C₇	semidirect
63. #389	C₂×C₇	semidirect
64. #384	C₂×C₇	semidirect
65. #380	C₂×C₇	semidirect
137. C₄×C₇	C₂²⋊Q₈	semidirect
141. C₂×Q₈	C₇×D₄	semidirect
144. C₂×D₄	C₇×Q₈	semidirect
207. C₇	#2538	direct (N × H)
228. 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×C₂²⋊Q₈	2
C₂×C₄×C₇×Q₈	2
C₂×C₄×C₇×D₄	2
C₇×#396	2
C₇×#390	2
C₇×#389	2
C₇×#384	2
C₇×#380	2
#2538	7

Subgroups — C7×#2538 #11741847

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2538 #11741847