Subgroups — C₇×#2507 #11741816

Order 896.

396 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#398	C₂	1	2	yes	maximal
2	C₇×#398	C₂	1	2	yes	maximal
3	C₇×#389	C₂	1	2	yes	maximal
4	C₇×#389	C₂	1	1	yes	maximal, characteristic
5	C₇×#389	C₂	1	1	yes	maximal, characteristic
6	C₇×#389	C₂	1	1	yes	maximal, characteristic
7	C₇×#389	C₂	1	1	yes	maximal, characteristic
8	C₇×#387	C₂	1	2	yes	maximal
9	C₇×#384	C₂	1	1	yes	maximal, characteristic
10	C₄×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
11	C₄×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
12	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
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×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×C₄²×C₇	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
26	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₇	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	C₇×#95	C₂²	1	2	yes	—
33	C₇×#95	C₂²	1	2	yes	—
34	C₇×#95	C₂²	1	2	yes	—
35	#2507	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₇	C₂³	1	2	yes	—
38	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₇	C₂³	1	2	yes	—
40	C₂²×C₄×C₇	C₂³	1	2	yes	—
41	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₇	C₂³	1	2	yes	—
45	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
47	C₇×C₄⋊C₄	—	2	2	no	—
48	C₇×C₄⋊C₄	—	2	4	no	—
49	C₇×C₄⋊C₄	—	2	2	no	—
50	C₇×C₄⋊C₄	—	2	2	no	—
51	C₇×C₄⋊C₄	—	2	2	no	—
52	C₇×C₄⋊C₄	—	4	2	no	—
53	C₇×C₄⋊C₄	—	4	2	no	—
54	C₇×C₄⋊C₄	—	4	2	no	—
55	C₇×C₄⋊C₄	—	4	2	no	—
56	C₇×C₄⋊C₄	—	4	2	no	—
57	C₇×C₄⋊C₄	Q₈	1	4	yes	—
58	C₇×C₄⋊C₄	Q₈	1	4	yes	—
59	C₄²×C₇	—	2	2	no	—
60	C₄²×C₇	—	2	2	no	—
61	C₄²×C₇	—	2	2	no	—
62	C₄²×C₇	—	4	2	no	—
63	#398	C₂×C₇	1	2	yes	—
64	#398	C₂×C₇	1	2	yes	—
65	#389	C₂×C₇	1	2	yes	—
66	#389	C₂×C₇	1	1	yes	characteristic
67	#389	C₂×C₇	1	1	yes	characteristic
68	#389	C₂×C₇	1	1	yes	characteristic
69	#389	C₂×C₇	1	1	yes	characteristic
70	#387	C₂×C₇	1	2	yes	—
71	#384	C₂×C₇	1	1	yes	characteristic
72	C₄×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic
73	C₄×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic
74	C₂³×C₇	C₂⁴	1	1	yes	characteristic
75	C₂×C₄×C₇	—	2	1	no	—
76	C₂×C₄×C₇	—	2	1	no	—
77	C₂×C₄×C₇	—	2	1	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	1	no	—
82	C₂×C₄×C₇	—	2	2	no	—
83	C₂×C₄×C₇	—	2	1	no	—
84	C₂×C₄×C₇	—	2	1	no	—
85	C₂×C₄×C₇	—	2	1	no	—
86	C₂×C₄×C₇	—	2	1	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	1	no	—
91	C₂×C₄×C₇	—	2	2	no	—
92	C₂×C₄×C₇	—	2	2	no	—
93	C₂×C₄×C₇	—	2	1	no	—
94	C₂×C₄×C₇	—	2	2	no	—
95	C₂×C₄×C₇	—	2	2	no	—
96	C₂×C₄×C₇	—	2	1	no	—
97	C₂×C₄×C₇	—	2	2	no	—
98	C₂×C₄×C₇	—	2	1	no	—
99	C₂×C₄×C₇	—	2	2	no	—
100	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
101	C₂×C₄×C₇	C₂×Q₈	1	4	yes	—
102	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
103	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
104	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
105	C₂×C₄×C₇	#41	1	2	yes	—
106	C₂×C₄×C₇	#41	1	2	yes	—
107	C₂×C₄×C₇	#41	1	2	yes	—
108	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
109	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
110	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
111	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
112	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
113	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
114	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
115	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
116	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
117	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
118	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
119	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
120	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
121	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
122	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
123	C₂×C₄²	C₂²×C₇	1	2	yes	—
124	#95	C₂²×C₇	1	1	yes	characteristic
125	#95	C₂²×C₇	1	1	yes	characteristic
126	#95	C₂²×C₇	1	2	yes	—
127	#95	C₂²×C₇	1	2	yes	—
128	#95	C₂²×C₇	1	2	yes	—
129	#95	C₂²×C₇	1	2	yes	—
130	#95	C₂²×C₇	1	2	yes	—
131	C₂²×C₇	C₂²×Q₈	1	1	yes	characteristic
132	C₂²×C₇	C₂²×Q₈	1	1	yes	characteristic
133	C₂²×C₇	C₂×#41	1	1	yes	characteristic
134	C₂²×C₇	C₂×#41	1	1	yes	characteristic
135	C₂²×C₇	C₂×#41	1	1	yes	characteristic
136	C₂²×C₇	H(5,𝔽₂)	1	1	yes	characteristic
137	C₂²×C₇	H(5,𝔽₂)	1	1	yes	characteristic
138	C₄×C₇	—	2	2	no	—
139	C₄×C₇	—	2	2	no	—
140	C₄×C₇	—	2	2	no	—
141	C₄×C₇	—	2	2	no	—
142	C₄×C₇	—	2	4	no	—
143	C₄×C₇	—	4	1	no	—
144	C₄×C₇	—	4	2	no	—
145	C₄×C₇	—	4	2	no	—
146	C₄×C₇	—	4	1	no	—
147	C₄×C₇	—	4	2	no	—
148	C₄×C₇	#125	1	4	yes	—
149	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
150	C₂²×C₄	C₂³×C₇	1	2	yes	—
151	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
152	C₂²×C₄	C₂³×C₇	1	2	yes	—
153	C₂²×C₄	C₂³×C₇	1	2	yes	—
154	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
155	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
156	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
157	C₂²×C₄	C₂³×C₇	1	2	yes	—
158	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
159	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
160	C₄⋊C₄	—	2	2	no	—
161	C₄⋊C₄	—	2	4	no	—
162	C₄⋊C₄	—	2	2	no	—
163	C₄⋊C₄	—	2	2	no	—
164	C₄⋊C₄	—	2	2	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄⋊C₄	—	4	2	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₄⋊C₄	C₇×Q₈	1	4	yes	—
171	C₄⋊C₄	C₇×Q₈	1	4	yes	—
172	C₄²	—	2	2	no	—
173	C₄²	—	2	2	no	—
174	C₄²	—	2	2	no	—
175	C₄²	—	4	2	no	—
176	C₂×C₇	C₂×#125	1	1	yes	characteristic
177	C₂×C₇	#533	1	1	yes	characteristic
178	C₂×C₇	#540	1	1	yes	characteristic
179	C₂×C₇	#553	1	1	yes	characteristic
180	C₂×C₇	#554	1	1	yes	characteristic
181	C₂×C₇	#554	1	1	yes	characteristic
182	C₂×C₇	Q₈²	1	1	yes	characteristic
183	C₂³	C₂⁴×C₇	1	1	yes	characteristic
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	C₂×C₇×Q₈	1	2	yes	—
210	C₂×C₄	C₂×C₇×Q₈	1	4	yes	—
211	C₂×C₄	C₂×C₇×Q₈	1	2	yes	—
212	C₂×C₄	C₂×C₇×Q₈	1	2	yes	—
213	C₂×C₄	C₂×C₇×Q₈	1	2	yes	—
214	C₂×C₄	C₇×#41	1	2	yes	—
215	C₂×C₄	C₇×#41	1	2	yes	—
216	C₂×C₄	C₇×#41	1	2	yes	—
217	C₇	#2507	1	1	yes	characteristic, direct product factor, Sylow
218	C₂²	C₂²×C₇×Q₈	1	1	yes	characteristic
219	C₂²	C₂²×C₇×Q₈	1	1	yes	characteristic
220	C₂²	C₂×C₇×#41	1	1	yes	characteristic
221	C₂²	C₂×C₇×#41	1	1	yes	characteristic
222	C₂²	C₂×C₇×#41	1	1	yes	characteristic
223	C₂²	C₇×H(5,𝔽₂)	1	1	yes	characteristic
224	C₂²	C₇×H(5,𝔽₂)	1	1	yes	characteristic
225	C₄	—	2	2	no	—
226	C₄	—	2	2	no	—
227	C₄	—	2	2	no	—
228	C₄	—	2	2	no	—
229	C₄	—	2	4	no	—
230	C₄	—	4	1	no	—
231	C₄	—	4	2	no	—
232	C₄	—	4	2	no	—
233	C₄	—	4	1	no	—
234	C₄	—	4	2	no	—
235	C₄	C₇×#125	1	4	yes	—
236	C₂	C₂×C₇×#125	1	1	yes	characteristic
237	C₂	C₇×#533	1	1	yes	characteristic
238	C₂	C₇×#540	1	1	yes	characteristic
239	C₂	C₇×#553	1	1	yes	characteristic
240	C₂	C₇×#554	1	1	yes	characteristic
241	C₂	C₇×#554	1	1	yes	characteristic
242	C₂	C₇×Q₈²	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₇
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
35. #2507	C₇	direct (N × H)
217. C₇	#2507	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#398	2
C₇×#389	2
C₇×#387	2
C₇×#384	2
C₄×C₇×C₄⋊C₄	2
#2507	7

Subgroups — C7×#2507 #11741816

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2507 #11741816