Subgroups — C₇×#22024 #13199240

Order 1792.

686 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₇×#1985	C₂	1	2	yes	maximal
3	C₇×#1940	C₂	1	2	yes	maximal
4	C₇×#1917	C₂	1	4	yes	maximal
5	C₇×#1839	C₂	1	2	yes	maximal
6	C₇×#1831	C₂	1	2	yes	maximal
7	C₇×#1767	C₂	1	1	yes	maximal, characteristic
8	C₇×#1745	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×C₇×Q₈	C₄	1	2	yes	—
10	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₇×C₄⋊C₈	C₂²	1	4	yes	—
13	C₂×C₇×#97	C₂²	1	2	yes	—
14	C₂×C₇×#97	C₂²	1	2	yes	—
15	C₄×C₇×C₄⋊C₄	C₄	1	2	yes	—
16	C₄×C₇×C₄⋊C₄	C₄	1	2	yes	—
17	C₄×C₇×C₄⋊C₄	C₄	1	2	yes	—
18	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
19	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₄³×C₇	C₂²	1	1	yes	characteristic
21	C₄³×C₇	C₂²	1	2	yes	—
22	C₇×#336	C₂²	1	4	yes	—
23	C₇×#336	C₂²	1	4	yes	—
24	C₇×#336	C₂²	1	1	yes	characteristic
25	C₇×#336	C₂²	1	1	yes	characteristic
26	C₇×#336	C₂²	1	2	yes	—
27	C₇×#336	C₂²	1	2	yes	—
28	C₇×#336	C₂²	1	4	yes	—
29	#22024	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂²×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₇×C₈	C₂³	1	2	yes	—
32	C₂²×C₇×C₈	C₂³	1	2	yes	—
33	C₂²×C₇×C₈	C₂³	1	4	yes	—
34	C₄×C₇×Q₈	—	2	8	no	—
35	C₄×C₇×Q₈	—	2	4	no	—
36	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
37	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
38	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
39	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
40	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₇	C₂³	1	2	yes	—
49	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₇	C₂³	1	2	yes	—
51	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
52	C₇×C₄⋊C₈	—	2	4	no	—
53	C₇×C₄⋊C₈	—	4	4	no	—
54	C₇×#97	—	2	4	no	—
55	C₇×#97	—	4	2	no	—
56	C₇×#95	C₂×C₄	1	2	yes	—
57	C₇×#95	C₂×C₄	1	1	yes	characteristic
58	C₇×#95	C₂×C₄	1	2	yes	—
59	C₇×#95	C₂×C₄	1	1	yes	characteristic
60	C₄²×Q₈	C₂×C₇	1	1	yes	characteristic
61	#1985	C₂×C₇	1	2	yes	—
62	#1940	C₂×C₇	1	2	yes	—
63	#1917	C₂×C₇	1	4	yes	—
64	#1839	C₂×C₇	1	2	yes	—
65	#1831	C₂×C₇	1	2	yes	—
66	#1767	C₂×C₇	1	1	yes	characteristic
67	#1745	C₂×C₇	1	1	yes	characteristic
68	C₂×C₇×Q₈	—	2	2	no	—
69	C₂×C₇×Q₈	—	2	4	no	—
70	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
79	C₂×C₇×C₈	—	2	2	no	—
80	C₂×C₇×C₈	—	2	2	no	—
81	C₂×C₇×C₈	—	2	4	no	—
82	C₂×C₇×C₈	—	2	4	no	—
83	C₂×C₇×C₈	—	2	2	no	—
84	C₂×C₇×C₈	—	2	2	no	—
85	C₂×C₇×C₈	—	2	2	no	—
86	C₂×C₇×C₈	—	2	2	no	—
87	C₂×C₇×C₈	—	2	4	no	—
88	C₇×C₄⋊C₄	—	2	4	no	—
89	C₇×C₄⋊C₄	—	2	2	no	—
90	C₇×C₄⋊C₄	—	2	4	no	—
91	C₇×C₄⋊C₄	—	2	4	no	—
92	C₇×C₄⋊C₄	—	2	4	no	—
93	C₄²×C₇	—	2	4	no	—
94	C₄²×C₇	—	2	4	no	—
95	C₄²×C₇	—	2	4	no	—
96	C₄²×C₇	—	2	1	no	—
97	C₄²×C₇	—	2	2	no	—
98	C₄²×C₇	—	2	2	no	—
99	C₄²×C₇	—	2	2	no	—
100	C₄²×C₇	—	2	1	no	—
101	C₄²×C₇	—	4	2	no	—
102	C₄²×C₇	—	4	2	no	—
103	C₄²×C₇	—	4	2	no	—
104	C₄²×C₇	M₁₆	1	8	yes	—
105	C₄²×C₇	#41	1	4	yes	—
106	C₂×C₄×Q₈	C₄×C₇	1	2	yes	—
107	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
108	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
109	C₂×C₄⋊C₈	C₂²×C₇	1	4	yes	—
110	C₂×#97	C₂²×C₇	1	2	yes	—
111	C₂×#97	C₂²×C₇	1	2	yes	—
112	C₄×C₄⋊C₄	C₄×C₇	1	2	yes	—
113	C₄×C₄⋊C₄	C₄×C₇	1	2	yes	—
114	C₄×C₄⋊C₄	C₄×C₇	1	2	yes	—
115	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
116	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
117	C₄³	C₂²×C₇	1	2	yes	—
118	C₄³	C₂²×C₇	1	1	yes	characteristic
119	#336	C₂²×C₇	1	1	yes	characteristic
120	#336	C₂²×C₇	1	2	yes	—
121	#336	C₂²×C₇	1	4	yes	—
122	#336	C₂²×C₇	1	1	yes	characteristic
123	#336	C₂²×C₇	1	4	yes	—
124	#336	C₂²×C₇	1	4	yes	—
125	#336	C₂²×C₇	1	2	yes	—
126	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
127	C₇×Q₈	—	2	8	no	—
128	C₂×C₄×C₇	—	2	2	no	—
129	C₂×C₄×C₇	—	2	2	no	—
130	C₂×C₄×C₇	—	2	2	no	—
131	C₂×C₄×C₇	—	2	2	no	—
132	C₂×C₄×C₇	—	2	2	no	—
133	C₂×C₄×C₇	—	2	2	no	—
134	C₂×C₄×C₇	—	2	2	no	—
135	C₂×C₄×C₇	—	2	2	no	—
136	C₂×C₄×C₇	—	2	2	no	—
137	C₂×C₄×C₇	—	2	2	no	—
138	C₂×C₄×C₇	—	2	2	no	—
139	C₂×C₄×C₇	—	2	2	no	—
140	C₂×C₄×C₇	—	2	2	no	—
141	C₂×C₄×C₇	—	2	2	no	—
142	C₂×C₄×C₇	—	2	2	no	—
143	C₂×C₄×C₇	—	2	2	no	—
144	C₂×C₄×C₇	—	2	2	no	—
145	C₂×C₄×C₇	#117	1	2	yes	—
146	C₂×C₄×C₇	#117	1	2	yes	—
147	C₂×C₄×C₇	C₂×M₁₆	1	4	yes	—
148	C₂×C₄×C₇	C₂×M₁₆	1	4	yes	—
149	C₂×C₄×C₇	C₂×M₁₆	1	4	yes	—
150	C₂×C₄×C₇	C₂×#41	1	2	yes	—
151	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
152	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
153	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
154	C₇×C₈	—	4	2	no	—
155	C₇×C₈	—	4	2	no	—
156	C₇×C₈	—	4	4	no	—
157	C₂²×Q₈	C₂×C₄×C₇	1	1	yes	characteristic
158	C₂²×C₈	C₂³×C₇	1	2	yes	—
159	C₂²×C₈	C₂³×C₇	1	2	yes	—
160	C₂²×C₈	C₂³×C₇	1	4	yes	—
161	C₄×Q₈	—	2	8	no	—
162	C₄×Q₈	—	2	4	no	—
163	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
164	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
165	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
166	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
167	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
168	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
169	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
170	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
171	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
172	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
173	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
174	C₂×C₄²	C₂³×C₇	1	2	yes	—
175	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
176	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
177	C₂×C₄²	C₂³×C₇	1	2	yes	—
178	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
179	C₄⋊C₈	—	2	4	no	—
180	C₄⋊C₈	—	4	4	no	—
181	#97	—	2	4	no	—
182	#97	—	4	2	no	—
183	#95	C₂×C₄×C₇	1	2	yes	—
184	#95	C₂×C₄×C₇	1	1	yes	characteristic
185	#95	C₂×C₄×C₇	1	2	yes	—
186	#95	C₂×C₄×C₇	1	1	yes	characteristic
187	C₂²×C₇	C₂×#117	1	1	yes	characteristic
188	C₂²×C₇	#519	1	1	yes	characteristic
189	C₂²×C₇	#520	1	1	yes	characteristic
190	C₂²×C₇	C₂²×M₁₆	1	2	yes	—
191	C₂²×C₇	#568	1	2	yes	—
192	C₄×C₇	—	2	2	no	—
193	C₄×C₇	—	2	4	no	—
194	C₄×C₇	—	2	2	no	—
195	C₄×C₇	—	2	4	no	—
196	C₄×C₇	—	2	4	no	—
197	C₄×C₇	—	4	2	no	—
198	C₄×C₇	—	4	2	no	—
199	C₄×C₇	—	4	2	no	—
200	C₄×C₇	#541	1	2	yes	—
201	C₄×C₇	#542	1	2	yes	—
202	C₂×Q₈	—	2	2	no	—
203	C₂×Q₈	—	2	4	no	—
204	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
205	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
206	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
207	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
208	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
209	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
210	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
211	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
212	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	4	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄²	—	2	4	no	—
228	C₄²	—	2	4	no	—
229	C₄²	—	2	4	no	—
230	C₄²	—	2	1	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	2	2	no	—
234	C₄²	—	2	1	no	—
235	C₄²	—	4	2	no	—
236	C₄²	—	4	2	no	—
237	C₄²	—	4	2	no	—
238	C₄²	C₇×M₁₆	1	8	yes	—
239	C₄²	C₇×#41	1	4	yes	—
240	C₂×C₇	#2320	1	1	yes	characteristic
241	C₂×C₇	#2920	1	1	yes	characteristic
242	C₂×C₇	#2923	1	1	yes	characteristic
243	C₂×C₇	#2983	1	2	yes	—
244	C₂×C₇	#2989	1	2	yes	—
245	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
246	Q₈	—	2	8	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	C₇×#117	1	2	yes	—
265	C₂×C₄	C₇×#117	1	2	yes	—
266	C₂×C₄	C₂×C₇×M₁₆	1	4	yes	—
267	C₂×C₄	C₂×C₇×M₁₆	1	4	yes	—
268	C₂×C₄	C₂×C₇×M₁₆	1	4	yes	—
269	C₂×C₄	C₂×C₇×#41	1	2	yes	—
270	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
271	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
272	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
273	C₈	—	4	2	no	—
274	C₈	—	4	2	no	—
275	C₈	—	4	4	no	—
276	C₇	#22024	1	1	yes	characteristic, direct product factor, Sylow
277	C₂²	C₂×C₇×#117	1	1	yes	characteristic
278	C₂²	C₇×#519	1	1	yes	characteristic
279	C₂²	C₇×#520	1	1	yes	characteristic
280	C₂²	C₂²×C₇×M₁₆	1	2	yes	—
281	C₂²	C₇×#568	1	2	yes	—
282	C₄	—	2	2	no	—
283	C₄	—	2	4	no	—
284	C₄	—	2	2	no	—
285	C₄	—	2	4	no	—
286	C₄	—	2	4	no	—
287	C₄	—	4	2	no	—
288	C₄	—	4	2	no	—
289	C₄	—	4	2	no	—
290	C₄	C₇×#541	1	2	yes	—
291	C₄	C₇×#542	1	2	yes	—
292	C₂	C₇×#2320	1	1	yes	characteristic
293	C₂	C₇×#2920	1	1	yes	characteristic
294	C₂	C₇×#2923	1	1	yes	characteristic
295	C₂	C₇×#2983	1	2	yes	—
296	C₂	C₇×#2989	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
29. #22024	C₇	direct (N × H)
276. C₇	#22024	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄²×C₇×Q₈	2
C₇×#1985	2
C₇×#1940	2
C₇×#1917	2
C₇×#1839	2
C₇×#1831	2
C₇×#1767	2
C₇×#1745	2
#22024	7

Subgroups — C7×#22024 #13199240

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22024 #13199240