Subgroups — C₇×#2549 #11741858

Order 896.

544 conjugacy classes of subgroups.

The subgroup lattice has 230 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
2	C₇×#398	C₂	1	2	yes	maximal, complemented
3	C₇×#395	C₂	1	2	yes	maximal, complemented
4	C₇×#391	C₂	1	2	yes	maximal, complemented
5	C₇×#389	C₂	1	2	yes	maximal, complemented
6	C₇×#385	C₂	1	2	yes	maximal
7	C₇×#380	C₂	1	1	yes	maximal, characteristic
8	C₄×C₇×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
9	C₂×C₇×#95	C₂	1	1	yes	maximal, characteristic
10	C₂²×C₇×Q₈	C₂²	1	2	yes	—
11	C₂³×C₄×C₇	C₂²	1	2	yes	—
12	C₂³×C₄×C₇	C₂²	1	1	yes	characteristic
13	C₇×C₂²⋊Q₈	—	2	8	no	—
14	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
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×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₄²×C₇	C₂²	1	2	yes	—
24	C₇×#95	C₂²	1	2	yes	—
25	C₇×#95	C₂²	1	1	yes	characteristic
26	C₇×#95	C₂²	1	1	yes	characteristic
27	C₇×#95	C₂²	1	2	yes	—
28	C₇×#95	C₂²	1	1	yes	characteristic
29	C₇×#95	C₂²	1	1	yes	characteristic
30	C₇×#95	C₂²	1	2	yes	—
31	C₇×#95	C₂²	1	2	yes	—
32	#2549	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂⁴×C₇	C₂³	1	1	yes	characteristic
34	C₂×C₇×Q₈	—	2	4	no	—
35	C₂×C₇×Q₈	—	2	4	no	—
36	C₂×C₇×Q₈	—	4	2	no	—
37	C₂²×C₄×C₇	—	2	4	no	—
38	C₂²×C₄×C₇	—	2	4	no	—
39	C₂²×C₄×C₇	—	4	2	no	—
40	C₂²×C₄×C₇	—	4	2	no	—
41	C₂²×C₄×C₇	Q₈	1	4	yes	complemented
42	C₂²×C₄×C₇	C₂³	1	2	yes	—
43	C₂²×C₄×C₇	C₂³	1	2	yes	—
44	C₂²×C₄×C₇	C₂³	1	2	yes	—
45	C₂²×C₄×C₇	C₂³	1	2	yes	—
46	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₇	C₂³	1	2	yes	—
49	C₂²×C₄×C₇	C₂³	1	2	yes	—
50	C₇×C₄⋊C₄	—	2	4	no	—
51	C₇×C₄⋊C₄	—	2	4	no	—
52	C₇×C₄⋊C₄	—	2	4	no	—
53	C₇×C₄⋊C₄	—	2	4	no	—
54	C₇×C₄⋊C₄	—	4	2	no	—
55	C₇×C₄⋊C₄	—	4	2	no	—
56	C₇×C₂²⋊C₄	—	2	4	no	—
57	C₇×C₂²⋊C₄	D₄	1	4	yes	—
58	C₇×C₂²⋊C₄	Q₈	1	4	yes	complemented
59	C₄²×C₇	—	4	2	no	—
60	C₂×C₂²⋊Q₈	C₂×C₇	1	2	yes	—
61	#398	C₂×C₇	1	2	yes	complemented
62	#395	C₂×C₇	1	2	yes	complemented
63	#391	C₂×C₇	1	2	yes	complemented
64	#389	C₂×C₇	1	2	yes	complemented
65	#385	C₂×C₇	1	2	yes	—
66	#380	C₂×C₇	1	1	yes	characteristic
67	C₄×C₂²⋊C₄	C₂×C₇	1	1	yes	characteristic
68	C₂×#95	C₂×C₇	1	1	yes	characteristic
69	C₂³×C₇	—	2	2	no	—
70	C₂³×C₇	—	2	2	no	—
71	C₂³×C₇	C₂×D₄	1	2	yes	—
72	C₂³×C₇	C₂×Q₈	1	2	yes	—
73	C₂³×C₇	C₂×Q₈	1	2	yes	—
74	C₂³×C₇	C₂⁴	1	1	yes	characteristic
75	C₇×Q₈	—	4	8	no	—
76	C₂×C₄×C₇	—	2	2	no	—
77	C₂×C₄×C₇	—	2	2	no	—
78	C₂×C₄×C₇	—	2	2	no	—
79	C₂×C₄×C₇	—	2	2	no	—
80	C₂×C₄×C₇	—	2	2	no	—
81	C₂×C₄×C₇	—	2	2	no	—
82	C₂×C₄×C₇	—	2	2	no	—
83	C₂×C₄×C₇	—	2	2	no	—
84	C₂×C₄×C₇	—	2	2	no	—
85	C₂×C₄×C₇	—	2	2	no	—
86	C₂×C₄×C₇	—	2	2	no	—
87	C₂×C₄×C₇	—	2	2	no	—
88	C₂×C₄×C₇	—	2	2	no	—
89	C₂×C₄×C₇	—	2	2	no	—
90	C₂×C₄×C₇	—	2	2	no	—
91	C₂×C₄×C₇	—	2	2	no	—
92	C₂×C₄×C₇	—	2	2	no	—
93	C₂×C₄×C₇	—	4	4	no	—
94	C₂×C₄×C₇	—	4	8	no	—
95	C₂×C₄×C₇	C₂×D₄	1	4	yes	—
96	C₂×C₄×C₇	C₂×Q₈	1	4	yes	—
97	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
98	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
99	C₂×C₄×C₇	#41	1	4	yes	—
100	C₂²×Q₈	C₂²×C₇	1	2	yes	—
101	C₂³×C₄	C₂²×C₇	1	2	yes	—
102	C₂³×C₄	C₂²×C₇	1	1	yes	characteristic
103	C₂²⋊Q₈	—	2	8	no	—
104	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
105	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
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	2	yes	—
109	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
110	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
111	C₂×C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic
112	C₂×C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic
113	C₂×C₄²	C₂²×C₇	1	2	yes	—
114	#95	C₂²×C₇	1	2	yes	—
115	#95	C₂²×C₇	1	1	yes	characteristic
116	#95	C₂²×C₇	1	1	yes	characteristic
117	#95	C₂²×C₇	1	2	yes	—
118	#95	C₂²×C₇	1	1	yes	characteristic
119	#95	C₂²×C₇	1	1	yes	characteristic
120	#95	C₂²×C₇	1	2	yes	—
121	#95	C₂²×C₇	1	2	yes	—
122	C₂²×C₇	—	2	2	no	—
123	C₂²×C₇	—	2	2	no	—
124	C₂²×C₇	—	2	4	no	—
125	C₂²×C₇	—	2	4	no	—
126	C₂²×C₇	C₄⋊Q₈	1	4	yes	—
127	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
128	C₂²×C₇	C₂²×Q₈	1	1	yes	characteristic
129	C₂²×C₇	C₂²×Q₈	1	1	yes	characteristic
130	C₂²×C₇	C₂×#41	1	2	yes	—
131	C₂²×C₇	H(5,𝔽₂)	1	2	yes	—
132	C₄×C₇	—	2	2	no	—
133	C₄×C₇	—	2	2	no	—
134	C₄×C₇	—	2	4	no	—
135	C₄×C₇	—	2	4	no	—
136	C₄×C₇	—	2	4	no	—
137	C₄×C₇	—	4	2	no	—
138	C₄×C₇	—	4	2	no	—
139	C₄×C₇	—	4	2	no	—
140	C₂⁴	C₂³×C₇	1	1	yes	characteristic
141	C₂×Q₈	—	2	4	no	—
142	C₂×Q₈	—	2	4	no	—
143	C₂×Q₈	—	4	2	no	—
144	C₂²×C₄	—	2	4	no	—
145	C₂²×C₄	—	2	4	no	—
146	C₂²×C₄	—	4	2	no	—
147	C₂²×C₄	—	4	2	no	—
148	C₂²×C₄	C₇×Q₈	1	4	yes	complemented
149	C₂²×C₄	C₂³×C₇	1	2	yes	—
150	C₂²×C₄	C₂³×C₇	1	2	yes	—
151	C₂²×C₄	C₂³×C₇	1	2	yes	—
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	C₂²×C₄	C₂³×C₇	1	2	yes	—
156	C₂²×C₄	C₂³×C₇	1	2	yes	—
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	2	4	no	—
159	C₄⋊C₄	—	2	4	no	—
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₂²⋊C₄	—	2	4	no	—
164	C₂²⋊C₄	C₇×D₄	1	4	yes	—
165	C₂²⋊C₄	C₇×Q₈	1	4	yes	complemented
166	C₄²	—	4	2	no	—
167	C₂×C₇	—	2	4	no	—
168	C₂×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
169	C₂×C₇	#533	1	1	yes	characteristic
170	C₂×C₇	#543	1	1	yes	characteristic
171	C₂×C₇	#546	1	2	yes	—
172	C₂×C₇	#554	1	2	yes	—
173	C₂³	—	2	2	no	—
174	C₂³	—	2	2	no	—
175	C₂³	C₂×C₇×D₄	1	2	yes	—
176	C₂³	C₂×C₇×Q₈	1	2	yes	—
177	C₂³	C₂×C₇×Q₈	1	2	yes	—
178	C₂³	C₂⁴×C₇	1	1	yes	characteristic
179	Q₈	—	4	8	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	4	4	no	—
198	C₂×C₄	—	4	8	no	—
199	C₂×C₄	C₂×C₇×D₄	1	4	yes	—
200	C₂×C₄	C₂×C₇×Q₈	1	4	yes	—
201	C₂×C₄	C₂×C₇×Q₈	1	2	yes	—
202	C₂×C₄	C₂×C₇×Q₈	1	2	yes	—
203	C₂×C₄	C₇×#41	1	4	yes	—
204	C₇	#2549	1	1	yes	characteristic, direct product factor, Sylow
205	C₂²	—	2	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	2	4	no	—
208	C₂²	—	2	4	no	—
209	C₂²	C₇×C₄⋊Q₈	1	4	yes	—
210	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
211	C₂²	C₂²×C₇×Q₈	1	1	yes	characteristic
212	C₂²	C₂²×C₇×Q₈	1	1	yes	characteristic
213	C₂²	C₂×C₇×#41	1	2	yes	—
214	C₂²	C₇×H(5,𝔽₂)	1	2	yes	—
215	C₄	—	2	2	no	—
216	C₄	—	2	2	no	—
217	C₄	—	2	4	no	—
218	C₄	—	2	4	no	—
219	C₄	—	2	4	no	—
220	C₄	—	4	2	no	—
221	C₄	—	4	2	no	—
222	C₄	—	4	2	no	—
223	C₂	—	2	4	no	—
224	C₂	C₂×C₇×C₄⋊Q₈	1	1	yes	characteristic
225	C₂	C₇×#533	1	1	yes	characteristic
226	C₂	C₇×#543	1	1	yes	characteristic
227	C₂	C₇×#546	1	2	yes	—
228	C₂	C₇×#554	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₇
prime radical	C₂⁴×C₇	C₂³
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
2. C₇×#398	C₂	semidirect
3. C₇×#395	C₂	semidirect
4. C₇×#391	C₂	semidirect
5. C₇×#389	C₂	semidirect
32. #2549	C₇	direct (N × H)
41. C₂²×C₄×C₇	Q₈	semidirect
58. C₇×C₂²⋊C₄	Q₈	semidirect
61. #398	C₂×C₇	semidirect
62. #395	C₂×C₇	semidirect
63. #391	C₂×C₇	semidirect
64. #389	C₂×C₇	semidirect
148. C₂²×C₄	C₇×Q₈	semidirect
165. C₂²⋊C₄	C₇×Q₈	semidirect
204. C₇	#2549	direct (N × H)

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
C₇×#398	2
C₇×#395	2
C₇×#391	2
C₇×#389	2
C₇×#385	2
C₇×#380	2
C₄×C₇×C₂²⋊C₄	2
C₂×C₇×#95	2
#2549	7

Subgroups — C7×#2549 #11741858

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2549 #11741858