Subgroups — C₇×#22747 #13199963

Order 1792.

622 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×Q₈×M₁₆	C₂	1	2	yes	maximal
2	C₂×C₇×#446	C₂	1	1	yes	maximal, characteristic
3	C₇×#1950	C₂	1	4	yes	maximal
4	C₇×#1950	C₂	1	2	yes	maximal
5	C₇×#1838	C₂	1	2	yes	maximal
6	C₇×#1837	C₂	1	1	yes	maximal, characteristic
7	C₇×#1778	C₂	1	2	yes	maximal
8	C₄×C₇×#108	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
10	C₇×#446	—	2	2	no	—
11	C₇×#446	—	2	4	no	—
12	C₇×#446	C₄	1	4	yes	—
13	C₇×#446	C₄	1	4	yes	—
14	C₇×C₈×Q₈	—	2	2	no	—
15	C₂×C₇×#108	C₂²	1	2	yes	—
16	C₂×C₇×#108	C₂²	1	4	yes	—
17	C₇×#423	C₂²	1	2	yes	—
18	C₇×#423	C₂²	1	4	yes	—
19	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
21	C₄×C₇×M₁₆	C₂²	1	2	yes	—
22	C₄×C₇×M₁₆	C₂²	1	4	yes	—
23	C₂×C₇×#97	C₂²	1	2	yes	—
24	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
25	C₇×#341	C₂²	1	1	yes	characteristic
26	C₇×#341	C₂²	1	1	yes	characteristic
27	C₇×#341	C₂²	1	2	yes	—
28	C₇×#341	C₂²	1	2	yes	—
29	C₇×#341	C₂²	1	4	yes	—
30	#22747	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂²×C₇×Q₈	Q₈	1	1	yes	characteristic
32	C₂×C₇×M₁₆	C₂³	1	2	yes	—
33	C₂×C₇×M₁₆	C₂³	1	4	yes	—
34	C₂×C₇×M₁₆	C₂³	1	2	yes	—
35	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
36	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
37	C₂²×C₇×C₈	C₂³	1	2	yes	—
38	C₄×C₇×Q₈	C₂×C₄	1	2	yes	—
39	C₄×C₇×Q₈	C₂×C₄	1	2	yes	—
40	C₄×C₇×Q₈	D₄	1	2	yes	—
41	C₄×C₇×Q₈	D₄	1	2	yes	—
42	C₂×C₇×C₄⋊C₄	Q₈	1	1	yes	characteristic
43	C₂×C₇×C₄⋊C₄	Q₈	1	2	yes	—
44	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₇	C₂³	1	2	yes	—
46	C₇×#108	—	2	4	no	—
47	C₇×#108	—	2	8	no	—
48	C₇×C₄⋊C₈	—	2	2	no	—
49	C₇×C₄⋊C₈	—	2	4	no	—
50	C₇×C₄⋊C₈	—	2	4	no	—
51	C₇×C₄⋊C₈	—	2	2	no	—
52	C₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
53	C₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
54	C₇×C₄⋊C₈	C₂×C₄	1	8	yes	—
55	C₇×#97	—	2	4	no	—
56	C₇×#97	—	2	2	no	—
57	C₇×#97	C₂×C₄	1	8	yes	—
58	C₄×C₇×C₈	—	2	2	no	—
59	C₄×C₇×C₈	—	2	4	no	—
60	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
61	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
62	Q₈×M₁₆	C₂×C₇	1	2	yes	—
63	C₂×#446	C₂×C₇	1	1	yes	characteristic
64	#1950	C₂×C₇	1	4	yes	—
65	#1950	C₂×C₇	1	2	yes	—
66	#1838	C₂×C₇	1	2	yes	—
67	#1837	C₂×C₇	1	1	yes	characteristic
68	#1778	C₂×C₇	1	2	yes	—
69	C₄×#108	C₂×C₇	1	1	yes	characteristic
70	C₂×C₇×Q₈	—	4	2	no	—
71	C₂×C₇×Q₈	C₄⋊C₄	1	2	yes	—
72	C₂×C₇×Q₈	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
74	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
75	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
78	C₇×M₁₆	—	2	2	no	—
79	C₇×M₁₆	—	2	2	no	—
80	C₇×M₁₆	—	2	4	no	—
81	C₇×M₁₆	—	2	2	no	—
82	C₇×M₁₆	—	2	4	no	—
83	C₇×M₁₆	#41	1	4	yes	—
84	C₂×C₇×C₈	—	2	2	no	—
85	C₂×C₇×C₈	—	2	2	no	—
86	C₂×C₇×C₈	—	2	4	no	—
87	C₂×C₇×C₈	—	2	2	no	—
88	C₂×C₇×C₈	—	2	2	no	—
89	C₂×C₇×C₈	—	2	4	no	—
90	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
91	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
92	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
93	C₇×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
94	C₇×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
95	C₇×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
96	C₇×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
97	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₇	C₂²×C₄	1	4	yes	—
100	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₇	C₂×D₄	1	4	yes	—
102	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
104	#446	—	2	2	no	—
105	#446	—	2	4	no	—
106	#446	C₄×C₇	1	4	yes	—
107	#446	C₄×C₇	1	4	yes	—
108	C₈×Q₈	—	2	2	no	—
109	C₂×#108	C₂²×C₇	1	2	yes	—
110	C₂×#108	C₂²×C₇	1	4	yes	—
111	#423	C₂²×C₇	1	2	yes	—
112	#423	C₂²×C₇	1	4	yes	—
113	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
114	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
115	C₄×M₁₆	C₂²×C₇	1	2	yes	—
116	C₄×M₁₆	C₂²×C₇	1	4	yes	—
117	C₂×#97	C₂²×C₇	1	2	yes	—
118	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
119	#341	C₂²×C₇	1	2	yes	—
120	#341	C₂²×C₇	1	2	yes	—
121	#341	C₂²×C₇	1	1	yes	characteristic
122	#341	C₂²×C₇	1	4	yes	—
123	#341	C₂²×C₇	1	1	yes	characteristic
124	C₂³×C₇	C₂²×Q₈	1	1	yes	characteristic
125	C₇×Q₈	—	2	4	no	—
126	C₇×Q₈	—	4	2	no	—
127	C₂×C₄×C₇	—	2	2	no	—
128	C₂×C₄×C₇	—	4	1	no	—
129	C₂×C₄×C₇	—	4	2	no	—
130	C₂×C₄×C₇	—	4	2	no	—
131	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
134	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
136	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₇	C₂×#41	1	2	yes	—
140	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
141	C₇×C₈	—	2	2	no	—
142	C₇×C₈	—	2	4	no	—
143	C₇×C₈	—	2	2	no	—
144	C₇×C₈	—	2	4	no	—
145	C₇×C₈	—	4	2	no	—
146	C₇×C₈	—	4	4	no	—
147	C₂²×Q₈	C₇×Q₈	1	1	yes	characteristic
148	C₂×M₁₆	C₂³×C₇	1	4	yes	—
149	C₂×M₁₆	C₂³×C₇	1	2	yes	—
150	C₂×M₁₆	C₂³×C₇	1	2	yes	—
151	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
152	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
153	C₂²×C₈	C₂³×C₇	1	2	yes	—
154	C₄×Q₈	C₂×C₄×C₇	1	2	yes	—
155	C₄×Q₈	C₂×C₄×C₇	1	2	yes	—
156	C₄×Q₈	C₇×D₄	1	2	yes	—
157	C₄×Q₈	C₇×D₄	1	2	yes	—
158	C₂×C₄⋊C₄	C₇×Q₈	1	1	yes	characteristic
159	C₂×C₄⋊C₄	C₇×Q₈	1	2	yes	—
160	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
161	C₂×C₄²	C₂³×C₇	1	2	yes	—
162	#108	—	2	4	no	—
163	#108	—	2	8	no	—
164	C₄⋊C₈	—	2	2	no	—
165	C₄⋊C₈	—	2	4	no	—
166	C₄⋊C₈	—	2	4	no	—
167	C₄⋊C₈	—	2	2	no	—
168	C₄⋊C₈	C₂×C₄×C₇	1	2	yes	—
169	C₄⋊C₈	C₂×C₄×C₇	1	2	yes	—
170	C₄⋊C₈	C₂×C₄×C₇	1	8	yes	—
171	#97	—	2	4	no	—
172	#97	—	2	2	no	—
173	#97	C₂×C₄×C₇	1	8	yes	—
174	C₄×C₈	—	2	2	no	—
175	C₄×C₈	—	2	4	no	—
176	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
177	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
178	C₂²×C₇	—	2	1	no	—
179	C₂²×C₇	—	2	1	no	—
180	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
181	C₂²×C₇	#557	1	2	yes	—
182	C₄×C₇	—	2	1	no	—
183	C₄×C₇	—	2	4	no	—
184	C₄×C₇	—	2	1	no	—
185	C₄×C₇	—	2	1	no	—
186	C₄×C₇	—	2	4	no	—
187	C₄×C₇	#430	1	2	yes	—
188	C₄×C₇	C₄×#41	1	1	yes	characteristic
189	C₄×C₇	#519	1	1	yes	characteristic
190	C₄×C₇	#548	1	2	yes	—
191	C₂×Q₈	—	4	2	no	—
192	C₂×Q₈	C₇×C₄⋊C₄	1	2	yes	—
193	C₂×Q₈	C₇×C₄⋊C₄	1	2	yes	—
194	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
197	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
198	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
199	M₁₆	—	2	2	no	—
200	M₁₆	—	2	2	no	—
201	M₁₆	—	2	4	no	—
202	M₁₆	—	2	4	no	—
203	M₁₆	—	2	2	no	—
204	M₁₆	C₇×#41	1	4	yes	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
212	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
213	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
214	C₄⋊C₄	C₇×C₄⋊C₄	1	4	yes	—
215	C₄⋊C₄	C₇×C₄⋊C₄	1	4	yes	—
216	C₄⋊C₄	C₇×C₄⋊C₄	1	2	yes	—
217	C₄⋊C₄	C₇×C₄⋊C₄	1	2	yes	—
218	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
219	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
220	C₄²	C₂²×C₄×C₇	1	4	yes	—
221	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
222	C₄²	C₂×C₇×D₄	1	4	yes	—
223	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
224	C₂×C₇	—	2	2	no	—
225	C₂×C₇	#2351	1	1	yes	characteristic
226	C₂×C₇	C₂×#430	1	1	yes	characteristic
227	C₂×C₇	#2916	1	1	yes	characteristic
228	C₂³	C₂²×C₇×Q₈	1	1	yes	characteristic
229	Q₈	—	2	4	no	—
230	Q₈	—	4	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
236	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
237	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
238	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
239	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
240	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
241	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
242	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
243	C₂×C₄	C₂×C₇×#41	1	2	yes	—
244	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
245	C₈	—	2	2	no	—
246	C₈	—	2	4	no	—
247	C₈	—	2	2	no	—
248	C₈	—	2	4	no	—
249	C₈	—	4	2	no	—
250	C₈	—	4	4	no	—
251	C₇	#22747	1	1	yes	characteristic, direct product factor, Sylow
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
255	C₂²	C₇×#557	1	2	yes	—
256	C₄	—	2	1	no	—
257	C₄	—	2	4	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	4	no	—
261	C₄	C₇×#430	1	2	yes	—
262	C₄	C₄×C₇×#41	1	1	yes	characteristic
263	C₄	C₇×#519	1	1	yes	characteristic
264	C₄	C₇×#548	1	2	yes	—
265	C₂	—	2	2	no	—
266	C₂	C₇×#2351	1	1	yes	characteristic
267	C₂	C₂×C₇×#430	1	1	yes	characteristic
268	C₂	C₇×#2916	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂²×C₇	C₂²×C₄⋊C₄
prime radical	C₂³×C₇	C₂²×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₄×C₇×Q₈ ◁ G

Complements

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

N	H ≅ G/N	type
30. #22747	C₇	direct (N × H)
251. C₇	#22747	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×Q₈×M₁₆	2
C₂×C₇×#446	2
C₇×#1950	2
C₇×#1838	2
C₇×#1837	2
C₇×#1778	2
C₄×C₇×#108	2
#22747	7

Subgroups — C7×#22747 #13199963

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22747 #13199963