Subgroups — C₇×#22733 #13199949

Order 1792.

638 conjugacy classes of subgroups.

The subgroup lattice has 292 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×#1849	C₂	1	2	yes	maximal
6	C₇×#1848	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	1	yes	characteristic
16	C₂×C₇×#108	C₂²	1	2	yes	—
17	C₂×C₇×#108	C₂²	1	2	yes	—
18	C₂×C₇×#108	C₂²	1	4	yes	—
19	C₂×C₇×#108	C₂²	1	1	yes	characteristic
20	C₇×#423	C₂²	1	2	yes	—
21	C₇×#423	C₂²	1	4	yes	—
22	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
24	C₄×C₇×M₁₆	C₂²	1	2	yes	—
25	C₄×C₇×M₁₆	C₂²	1	4	yes	—
26	C₂×C₇×#97	C₂²	1	2	yes	—
27	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
28	C₇×#341	C₂²	1	2	yes	—
29	C₇×#341	C₂²	1	4	yes	—
30	#22733	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	2	yes	—
37	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
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	2	no	—
47	C₇×#108	—	2	4	no	—
48	C₇×#108	—	4	2	no	—
49	C₇×#108	—	4	4	no	—
50	C₇×#108	Q₈	1	4	yes	complemented
51	C₇×C₄⋊C₈	—	2	2	no	—
52	C₇×C₄⋊C₈	—	2	4	no	—
53	C₇×C₄⋊C₈	—	2	4	no	—
54	C₇×C₄⋊C₈	—	2	2	no	—
55	C₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
56	C₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
57	C₇×C₄⋊C₈	C₂×C₄	1	8	yes	—
58	C₇×#97	—	2	4	no	—
59	C₇×#97	—	2	2	no	—
60	C₇×#97	C₂×C₄	1	8	yes	—
61	C₄×C₇×C₈	—	2	2	no	—
62	C₄×C₇×C₈	—	2	4	no	—
63	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
64	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
65	Q₈×M₁₆	C₂×C₇	1	2	yes	—
66	C₂×#446	C₂×C₇	1	1	yes	characteristic
67	#1950	C₂×C₇	1	4	yes	—
68	#1950	C₂×C₇	1	2	yes	—
69	#1849	C₂×C₇	1	2	yes	—
70	#1848	C₂×C₇	1	1	yes	characteristic
71	#1778	C₂×C₇	1	2	yes	—
72	C₄×#108	C₂×C₇	1	1	yes	characteristic
73	C₂×C₇×Q₈	—	4	2	no	—
74	C₂×C₇×Q₈	C₄⋊C₄	1	2	yes	—
75	C₂×C₇×Q₈	C₄⋊C₄	1	2	yes	—
76	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
77	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
78	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
79	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
80	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
81	C₇×M₁₆	—	2	2	no	—
82	C₇×M₁₆	—	2	2	no	—
83	C₇×M₁₆	—	2	4	no	—
84	C₇×M₁₆	—	2	4	no	—
85	C₇×M₁₆	—	2	2	no	—
86	C₇×M₁₆	C₂×Q₈	1	4	yes	—
87	C₂×C₇×C₈	—	2	2	no	—
88	C₂×C₇×C₈	—	2	1	no	—
89	C₂×C₇×C₈	—	2	2	no	—
90	C₂×C₇×C₈	—	2	1	no	—
91	C₂×C₇×C₈	—	2	2	no	—
92	C₂×C₇×C₈	—	2	2	no	—
93	C₂×C₇×C₈	—	2	4	no	—
94	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
95	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
96	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
97	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
98	C₂×C₇×C₈	#41	1	2	yes	—
99	C₇×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
100	C₇×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
101	C₇×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
102	C₇×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
103	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₇	C₂²×C₄	1	4	yes	—
106	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₇	C₂×D₄	1	4	yes	—
108	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
110	#446	—	2	2	no	—
111	#446	—	2	4	no	—
112	#446	C₄×C₇	1	4	yes	—
113	#446	C₄×C₇	1	4	yes	—
114	C₈×Q₈	—	2	2	no	—
115	C₂×#108	C₂²×C₇	1	2	yes	—
116	C₂×#108	C₂²×C₇	1	1	yes	characteristic
117	C₂×#108	C₂²×C₇	1	2	yes	—
118	C₂×#108	C₂²×C₇	1	4	yes	—
119	C₂×#108	C₂²×C₇	1	1	yes	characteristic
120	#423	C₂²×C₇	1	2	yes	—
121	#423	C₂²×C₇	1	4	yes	—
122	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
123	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
124	C₄×M₁₆	C₂²×C₇	1	2	yes	—
125	C₄×M₁₆	C₂²×C₇	1	4	yes	—
126	C₂×#97	C₂²×C₇	1	2	yes	—
127	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
128	#341	C₂²×C₇	1	2	yes	—
129	#341	C₂²×C₇	1	4	yes	—
130	C₂³×C₇	C₂²×Q₈	1	1	yes	characteristic
131	C₇×Q₈	—	2	4	no	—
132	C₇×Q₈	—	4	2	no	—
133	C₂×C₄×C₇	—	2	2	no	—
134	C₂×C₄×C₇	—	4	1	no	—
135	C₂×C₄×C₇	—	4	2	no	—
136	C₂×C₄×C₇	—	4	2	no	—
137	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
140	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
142	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
146	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
147	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
148	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
149	C₇×C₈	—	2	2	no	—
150	C₇×C₈	—	2	4	no	—
151	C₇×C₈	—	2	4	no	—
152	C₇×C₈	—	4	2	no	—
153	C₇×C₈	—	4	4	no	—
154	C₇×C₈	C₄×Q₈	1	4	yes	—
155	C₂²×Q₈	C₇×Q₈	1	1	yes	characteristic
156	C₂×M₁₆	C₂³×C₇	1	2	yes	—
157	C₂×M₁₆	C₂³×C₇	1	4	yes	—
158	C₂×M₁₆	C₂³×C₇	1	2	yes	—
159	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
160	C₂²×C₈	C₂³×C₇	1	2	yes	—
161	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
162	C₄×Q₈	C₂×C₄×C₇	1	2	yes	—
163	C₄×Q₈	C₂×C₄×C₇	1	2	yes	—
164	C₄×Q₈	C₇×D₄	1	2	yes	—
165	C₄×Q₈	C₇×D₄	1	2	yes	—
166	C₂×C₄⋊C₄	C₇×Q₈	1	1	yes	characteristic
167	C₂×C₄⋊C₄	C₇×Q₈	1	2	yes	—
168	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
169	C₂×C₄²	C₂³×C₇	1	2	yes	—
170	#108	—	2	4	no	—
171	#108	—	2	2	no	—
172	#108	—	4	2	no	—
173	#108	—	4	4	no	—
174	#108	C₇×Q₈	1	4	yes	complemented
175	C₄⋊C₈	—	2	2	no	—
176	C₄⋊C₈	—	2	4	no	—
177	C₄⋊C₈	—	2	4	no	—
178	C₄⋊C₈	—	2	2	no	—
179	C₄⋊C₈	C₂×C₄×C₇	1	2	yes	—
180	C₄⋊C₈	C₂×C₄×C₇	1	2	yes	—
181	C₄⋊C₈	C₂×C₄×C₇	1	8	yes	—
182	#97	—	2	4	no	—
183	#97	—	2	2	no	—
184	#97	C₂×C₄×C₇	1	8	yes	—
185	C₄×C₈	—	2	2	no	—
186	C₄×C₈	—	2	4	no	—
187	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
188	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
189	C₂²×C₇	—	2	1	no	—
190	C₂²×C₇	—	2	1	no	—
191	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
192	C₂²×C₇	#557	1	1	yes	characteristic
193	C₂²×C₇	Q₈²	1	1	yes	characteristic
194	C₄×C₇	—	2	1	no	—
195	C₄×C₇	—	2	4	no	—
196	C₄×C₇	—	2	1	no	—
197	C₄×C₇	—	2	1	no	—
198	C₄×C₇	—	2	4	no	—
199	C₄×C₇	#430	1	2	yes	—
200	C₄×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
201	C₄×C₇	#520	1	1	yes	characteristic
202	C₄×C₇	D₄×Q₈	1	1	yes	characteristic
203	C₄×C₇	#550	1	1	yes	characteristic
204	C₂×Q₈	—	4	2	no	—
205	C₂×Q₈	C₇×C₄⋊C₄	1	2	yes	—
206	C₂×Q₈	C₇×C₄⋊C₄	1	2	yes	—
207	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
210	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
211	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
212	M₁₆	—	2	2	no	—
213	M₁₆	—	2	4	no	—
214	M₁₆	—	2	2	no	—
215	M₁₆	—	2	4	no	—
216	M₁₆	—	2	2	no	—
217	M₁₆	C₂×C₇×Q₈	1	4	yes	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
226	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
227	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
228	C₂×C₈	C₂×C₇×Q₈	1	2	yes	—
229	C₂×C₈	C₇×#41	1	2	yes	—
230	C₄⋊C₄	C₇×C₄⋊C₄	1	4	yes	—
231	C₄⋊C₄	C₇×C₄⋊C₄	1	4	yes	—
232	C₄⋊C₄	C₇×C₄⋊C₄	1	2	yes	—
233	C₄⋊C₄	C₇×C₄⋊C₄	1	2	yes	—
234	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
235	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
236	C₄²	C₂²×C₄×C₇	1	4	yes	—
237	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
238	C₄²	C₂×C₇×D₄	1	4	yes	—
239	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
240	C₂×C₇	—	2	1	no	—
241	C₂×C₇	—	2	1	no	—
242	C₂×C₇	Q₈×C₄⋊C₄	1	1	yes	characteristic
243	C₂×C₇	C₂×#430	1	1	yes	characteristic
244	C₂×C₇	#2916	1	1	yes	characteristic
245	C₂³	C₂²×C₇×Q₈	1	1	yes	characteristic
246	Q₈	—	2	4	no	—
247	Q₈	—	4	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
253	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
254	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
255	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
256	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
257	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
258	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
260	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
261	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
262	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
263	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	2	4	no	—
266	C₈	—	2	4	no	—
267	C₈	—	4	2	no	—
268	C₈	—	4	4	no	—
269	C₈	C₄×C₇×Q₈	1	4	yes	—
270	C₇	#22733	1	1	yes	characteristic, direct product factor, Sylow
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
274	C₂²	C₇×#557	1	1	yes	characteristic
275	C₂²	C₇×Q₈²	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	4	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	4	no	—
281	C₄	C₇×#430	1	2	yes	—
282	C₄	C₂×C₄×C₇×Q₈	1	1	yes	characteristic
283	C₄	C₇×#520	1	1	yes	characteristic
284	C₄	C₇×D₄×Q₈	1	1	yes	characteristic
285	C₄	C₇×#550	1	1	yes	characteristic
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	C₇×Q₈×C₄⋊C₄	1	1	yes	characteristic
289	C₂	C₂×C₇×#430	1	1	yes	characteristic
290	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. #22733	C₇	direct (N × H)
50. C₇×#108	Q₈	semidirect
174. #108	C₇×Q₈	semidirect
270. C₇	#22733	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×#1849	2
C₇×#1848	2
C₇×#1778	2
C₄×C₇×#108	2
#22733	7

Subgroups — C7×#22733 #13199949

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22733 #13199949