Subgroups — C₇×#33423 #13210639

Order 1792.

590 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×#131	C₂	1	1	yes	maximal, characteristic
2	C₇×#2164	C₂	1	4	yes	maximal, complemented
3	C₇×#2163	C₂	1	2	yes	maximal, complemented
4	C₇×#2162	C₂	1	2	yes	maximal, complemented
5	C₇×#2150	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#2113	C₂	1	2	yes	maximal
8	C₇×#2108	C₂	1	2	yes	maximal, complemented
9	C₂×C₇×#131	C₄	1	1	yes	characteristic
10	C₂×C₇×#131	C₂²	1	1	yes	characteristic
11	C₄×C₇×#41	C₄	1	1	yes	characteristic
12	C₂×C₇×M₃₂	C₂²	1	2	yes	—
13	C₂²×C₇×C₁₆	C₂²	1	2	yes	—
14	C₇×#405	C₄	1	1	yes	characteristic
15	C₇×#405	C₄	1	2	yes	—
16	C₇×#405	C₂²	1	1	yes	characteristic
17	C₇×#405	C₂²	1	2	yes	—
18	C₄×C₇×M₁₆	C₄	1	2	yes	—
19	C₄×C₇×M₁₆	C₄	1	1	yes	characteristic
20	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
22	C₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
23	C₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
24	C₇×C₄⋊C₁₆	C₂²	1	2	yes	—
25	C₇×C₄⋊C₁₆	C₂²	1	2	yes	—
26	C₇×C₄⋊C₁₆	C₂²	1	2	yes	—
27	C₇×C₂²⋊C₁₆	C₂²	1	4	yes	—
28	C₇×C₂²⋊C₁₆	C₂²	1	4	yes	—
29	C₇×#346	C₂²	1	4	yes	—
30	C₄×C₇×C₁₆	C₂²	1	4	yes	—
31	#33423	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂×C₇×#41	C₂×C₄	1	1	yes	characteristic
33	C₇×#131	—	2	4	no	—
34	C₇×#131	—	2	2	no	—
35	C₇×#131	—	2	2	no	—
36	C₂×C₇×M₁₆	C₂×C₄	1	1	yes	characteristic
37	C₂×C₇×M₁₆	C₂×C₄	1	2	yes	—
38	C₂×C₇×M₁₆	C₂×C₄	1	2	yes	—
39	C₂×C₇×M₁₆	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
42	C₂²×C₇×C₈	C₂³	1	2	yes	—
43	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
44	C₄×C₇×Q₈	C₈	1	1	yes	characteristic
45	C₄×C₇×Q₈	C₈	1	1	yes	characteristic
46	C₄×C₇×D₄	C₈	1	2	yes	—
47	C₄×C₇×D₄	C₈	1	1	yes	characteristic
48	C₄×C₇×D₄	C₈	1	2	yes	—
49	C₄×C₇×D₄	C₈	1	1	yes	characteristic
50	C₇×#117	C₂×C₄	1	2	yes	—
51	C₇×#117	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
54	C₇×M₃₂	—	2	4	no	—
55	C₂×C₇×C₁₆	—	2	4	no	—
56	C₂×C₇×C₁₆	C₂³	1	2	yes	—
57	C₂×C₇×C₁₆	C₂³	1	2	yes	—
58	C₂×C₇×C₁₆	C₂³	1	4	yes	—
59	C₇×#97	C₂×C₄	1	2	yes	—
60	C₇×#97	C₂×C₄	1	2	yes	—
61	C₇×#97	C₂×C₄	1	2	yes	—
62	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
63	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
64	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
65	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
66	C₄×C₇×C₈	C₂³	1	2	yes	—
67	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
68	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
69	C₄×#131	C₂×C₇	1	1	yes	characteristic
70	#2164	C₂×C₇	1	4	yes	complemented
71	#2163	C₂×C₇	1	2	yes	complemented
72	#2162	C₂×C₇	1	2	yes	complemented
73	#2150	C₂×C₇	1	1	yes	characteristic, complemented
74	C₂×C₄⋊C₁₆	C₂×C₇	1	1	yes	characteristic, complemented
75	#2113	C₂×C₇	1	2	yes	—
76	#2108	C₂×C₇	1	2	yes	complemented
77	C₇×#41	—	2	2	no	—
78	C₇×#41	—	2	2	no	—
79	C₂×C₇×Q₈	C₂×C₈	1	1	yes	characteristic
80	C₂×C₇×D₄	C₂×C₈	1	2	yes	—
81	C₂×C₇×D₄	C₂×C₈	1	1	yes	characteristic
82	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
83	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
84	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
85	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
86	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
88	C₇×M₁₆	—	2	2	no	—
89	C₇×M₁₆	—	2	4	no	—
90	C₇×M₁₆	—	2	4	no	—
91	C₇×M₁₆	M₁₆	1	4	yes	—
92	C₂×C₇×C₈	—	2	2	no	—
93	C₂×C₇×C₈	—	2	4	no	—
94	C₂×C₇×C₈	—	2	2	no	—
95	C₂×C₇×C₈	—	2	2	no	—
96	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
98	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
101	C₂×C₇×C₈	#41	1	2	yes	—
102	C₂×C₇×C₈	#41	1	2	yes	—
103	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
104	C₇×C₄⋊C₄	C₂×C₈	1	2	yes	—
105	C₇×C₄⋊C₄	C₂×C₈	1	2	yes	—
106	C₇×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
107	C₇×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
108	C₇×C₂²⋊C₄	C₂×C₈	1	2	yes	—
109	C₇×C₂²⋊C₄	C₂×C₈	1	2	yes	—
110	C₇×C₂²⋊C₄	C₂×C₈	1	2	yes	—
111	C₄²×C₇	C₂×C₈	1	2	yes	—
112	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
113	C₄²×C₇	C₂×C₈	1	2	yes	—
114	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
115	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
116	C₄²×C₇	C₂²×C₄	1	2	yes	—
117	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
118	C₇×C₁₆	—	2	2	no	—
119	C₇×C₁₆	—	2	2	no	—
120	C₇×C₁₆	—	2	4	no	—
121	C₂×#131	C₄×C₇	1	1	yes	characteristic
122	C₂×#131	C₂²×C₇	1	1	yes	characteristic
123	C₄×#41	C₄×C₇	1	1	yes	characteristic
124	C₂×M₃₂	C₂²×C₇	1	2	yes	—
125	C₂²×C₁₆	C₂²×C₇	1	2	yes	—
126	#405	C₄×C₇	1	1	yes	characteristic
127	#405	C₄×C₇	1	2	yes	—
128	#405	C₂²×C₇	1	1	yes	characteristic
129	#405	C₂²×C₇	1	2	yes	—
130	C₄×M₁₆	C₄×C₇	1	2	yes	—
131	C₄×M₁₆	C₄×C₇	1	1	yes	characteristic
132	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
133	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
134	C₄⋊C₁₆	C₂²×C₇	1	1	yes	characteristic
135	C₄⋊C₁₆	C₂²×C₇	1	1	yes	characteristic
136	C₄⋊C₁₆	C₂²×C₇	1	2	yes	—
137	C₄⋊C₁₆	C₂²×C₇	1	2	yes	—
138	C₄⋊C₁₆	C₂²×C₇	1	2	yes	—
139	C₂²⋊C₁₆	C₂²×C₇	1	4	yes	—
140	C₂²⋊C₁₆	C₂²×C₇	1	4	yes	—
141	#346	C₂²×C₇	1	4	yes	—
142	C₄×C₁₆	C₂²×C₇	1	4	yes	—
143	C₂³×C₇	C₂²×C₈	1	2	yes	—
144	C₂³×C₇	C₂²×C₈	1	1	yes	characteristic
145	C₇×Q₈	—	2	2	no	—
146	C₇×D₄	—	2	4	no	—
147	C₇×D₄	—	2	2	no	—
148	C₂×C₄×C₇	—	2	2	no	—
149	C₂×C₄×C₇	—	2	2	no	—
150	C₂×C₄×C₇	—	2	2	no	—
151	C₂×C₄×C₇	—	2	4	no	—
152	C₂×C₄×C₇	—	4	1	no	—
153	C₂×C₄×C₇	—	4	2	no	—
154	C₂×C₄×C₇	#117	1	2	yes	—
155	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
156	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
157	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
158	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
159	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
160	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
161	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
162	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
163	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
164	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
165	C₇×C₈	—	2	2	no	—
166	C₇×C₈	—	2	1	no	—
167	C₇×C₈	—	2	1	no	—
168	C₇×C₈	#117	1	2	yes	—
169	C₇×C₈	C₂×M₁₆	1	4	yes	—
170	C₇×C₈	C₂×#41	1	1	yes	characteristic
171	C₇×C₈	C₂×#41	1	1	yes	characteristic
172	C₂×#41	C₂×C₄×C₇	1	1	yes	characteristic
173	#131	—	2	4	no	—
174	#131	—	2	2	no	—
175	#131	—	2	2	no	—
176	C₂×M₁₆	C₂×C₄×C₇	1	2	yes	—
177	C₂×M₁₆	C₂×C₄×C₇	1	1	yes	characteristic
178	C₂×M₁₆	C₂×C₄×C₇	1	2	yes	—
179	C₂×M₁₆	C₂×C₄×C₇	1	1	yes	characteristic
180	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
181	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
182	C₂²×C₈	C₂³×C₇	1	2	yes	—
183	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
184	C₄×Q₈	C₇×C₈	1	1	yes	characteristic
185	C₄×Q₈	C₇×C₈	1	1	yes	characteristic
186	C₄×D₄	C₇×C₈	1	2	yes	—
187	C₄×D₄	C₇×C₈	1	1	yes	characteristic
188	C₄×D₄	C₇×C₈	1	2	yes	—
189	C₄×D₄	C₇×C₈	1	1	yes	characteristic
190	#117	C₂×C₄×C₇	1	2	yes	—
191	#117	C₂×C₄×C₇	1	1	yes	characteristic
192	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
193	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
194	M₃₂	—	2	4	no	—
195	C₂×C₁₆	—	2	4	no	—
196	C₂×C₁₆	C₂³×C₇	1	2	yes	—
197	C₂×C₁₆	C₂³×C₇	1	4	yes	—
198	C₂×C₁₆	C₂³×C₇	1	2	yes	—
199	#97	C₂×C₄×C₇	1	2	yes	—
200	#97	C₂×C₄×C₇	1	2	yes	—
201	#97	C₂×C₄×C₇	1	2	yes	—
202	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
203	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
204	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
205	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
206	C₄×C₈	C₂³×C₇	1	2	yes	—
207	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
208	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
209	C₂²×C₇	—	2	1	no	—
210	C₂²×C₇	—	2	1	no	—
211	C₂²×C₇	—	2	2	no	—
212	C₂²×C₇	—	2	2	no	—
213	C₂²×C₇	—	2	2	no	—
214	C₂²×C₇	#431	1	2	yes	—
215	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
216	C₄×C₇	—	2	1	no	—
217	C₄×C₇	—	2	2	no	—
218	C₄×C₇	—	2	1	no	—
219	C₄×C₇	—	2	1	no	—
220	C₄×C₇	—	2	2	no	—
221	C₄×C₇	—	2	2	no	—
222	C₄×C₇	#431	1	2	yes	—
223	C₄×C₇	#504	1	2	yes	—
224	C₄×C₇	C₂×#117	1	1	yes	characteristic
225	C₄×C₇	C₂²×M₁₆	1	1	yes	characteristic
226	#41	—	2	2	no	—
227	#41	—	2	2	no	—
228	C₂×Q₈	C₂×C₇×C₈	1	1	yes	characteristic
229	C₂×D₄	C₂×C₇×C₈	1	2	yes	—
230	C₂×D₄	C₂×C₇×C₈	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₇×C₈	1	2	yes	—
232	C₂²×C₄	C₂×C₇×C₈	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₇×C₈	1	2	yes	—
234	C₂²×C₄	C₂×C₇×C₈	1	1	yes	characteristic
235	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
236	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
237	M₁₆	—	2	2	no	—
238	M₁₆	—	2	4	no	—
239	M₁₆	—	2	4	no	—
240	M₁₆	C₇×M₁₆	1	4	yes	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
246	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
247	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
248	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
249	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
250	C₂×C₈	C₇×#41	1	2	yes	—
251	C₂×C₈	C₇×#41	1	2	yes	—
252	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
253	C₄⋊C₄	C₂×C₇×C₈	1	2	yes	—
254	C₄⋊C₄	C₂×C₇×C₈	1	2	yes	—
255	C₄⋊C₄	C₂×C₇×C₈	1	1	yes	characteristic
256	C₄⋊C₄	C₂×C₇×C₈	1	1	yes	characteristic
257	C₂²⋊C₄	C₂×C₇×C₈	1	2	yes	—
258	C₂²⋊C₄	C₂×C₇×C₈	1	2	yes	—
259	C₂²⋊C₄	C₂×C₇×C₈	1	2	yes	—
260	C₄²	C₂×C₇×C₈	1	2	yes	—
261	C₄²	C₂×C₇×C₈	1	1	yes	characteristic
262	C₄²	C₂×C₇×C₈	1	2	yes	—
263	C₄²	C₂×C₇×C₈	1	1	yes	characteristic
264	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
265	C₄²	C₂²×C₄×C₇	1	2	yes	—
266	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
267	C₁₆	—	2	2	no	—
268	C₁₆	—	2	2	no	—
269	C₁₆	—	2	4	no	—
270	C₂×C₇	—	2	2	no	—
271	C₂×C₇	—	4	2	no	—
272	C₂×C₇	C₂×#431	1	1	yes	characteristic
273	C₂×C₇	C₂×#504	1	1	yes	characteristic
274	C₂×C₇	#3407	1	1	yes	characteristic
275	C₂³	C₂²×C₇×C₈	1	2	yes	—
276	C₂³	C₂²×C₇×C₈	1	1	yes	characteristic
277	Q₈	—	2	2	no	—
278	D₄	—	2	4	no	—
279	D₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	C₇×#117	1	2	yes	—
287	C₂×C₄	C₂²×C₇×C₈	1	1	yes	characteristic
288	C₂×C₄	C₂²×C₇×C₈	1	2	yes	—
289	C₂×C₄	C₂²×C₇×C₈	1	2	yes	—
290	C₂×C₄	C₂²×C₇×C₈	1	1	yes	characteristic
291	C₂×C₄	C₂²×C₇×C₈	1	2	yes	—
292	C₂×C₄	C₂²×C₇×C₈	1	1	yes	characteristic
293	C₂×C₄	C₂²×C₇×C₈	1	1	yes	characteristic
294	C₂×C₄	C₂²×C₇×C₈	1	1	yes	characteristic
295	C₂×C₄	C₂×C₇×M₁₆	1	2	yes	—
296	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
297	C₈	—	2	2	no	—
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	C₇×#117	1	2	yes	—
301	C₈	C₂×C₇×M₁₆	1	4	yes	—
302	C₈	C₂×C₇×#41	1	1	yes	characteristic
303	C₈	C₂×C₇×#41	1	1	yes	characteristic
304	C₇	#33423	1	1	yes	characteristic, direct product factor, Sylow
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	C₇×#431	1	2	yes	—
311	C₂²	C₂³×C₇×C₈	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	2	no	—
318	C₄	C₇×#431	1	2	yes	—
319	C₄	C₇×#504	1	2	yes	—
320	C₄	C₂×C₇×#117	1	1	yes	characteristic
321	C₄	C₂²×C₇×M₁₆	1	1	yes	characteristic
322	C₂	—	2	2	no	—
323	C₂	—	4	2	no	—
324	C₂	C₂×C₇×#431	1	1	yes	characteristic
325	C₂	C₂×C₇×#504	1	1	yes	characteristic
326	C₂	C₇×#3407	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₇×C₈	C₂⁴
commutator	C₂²	C₂³×C₇×C₈
Frattini, cosocle	C₂×C₈	C₂⁴×C₇
socle	C₂²×C₇	C₂³×C₈
prime radical	C₂×C₇×#41	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#2164	C₂	semidirect
3. C₇×#2163	C₂	semidirect
4. C₇×#2162	C₂	semidirect
5. C₇×#2150	C₂	semidirect
6. C₂×C₇×C₄⋊C₁₆	C₂	semidirect
8. C₇×#2108	C₂	semidirect
31. #33423	C₇	direct (N × H)
70. #2164	C₂×C₇	semidirect
71. #2163	C₂×C₇	semidirect
72. #2162	C₂×C₇	semidirect
73. #2150	C₂×C₇	semidirect
74. C₂×C₄⋊C₁₆	C₂×C₇	semidirect
76. #2108	C₂×C₇	semidirect
304. C₇	#33423	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₇×#131	2
C₇×#2164	2
C₇×#2163	2
C₇×#2162	2
C₇×#2150	2
C₂×C₇×C₄⋊C₁₆	2
C₇×#2113	2
C₇×#2108	2
#33423	7

Subgroups — C7×#33423 #13210639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#33423 #13210639