Subgroups — C₇×#33707 #13210923

Order 1792.

622 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#3147	C₂	1	1	yes	maximal, characteristic
3	C₇×#2243	C₂	1	4	yes	maximal, complemented
4	C₇×#2241	C₂	1	2	yes	maximal, complemented
5	C₇×#2240	C₂	1	2	yes	maximal, complemented
6	C₇×#2139	C₂	1	4	yes	maximal, complemented
7	C₄×C₇×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₇×#135	—	2	1	no	—
9	C₂×C₇×#135	C₂²	1	2	yes	—
10	C₇×#532	—	2	1	no	—
11	C₂×C₇×C₄⋊Q₈	—	2	1	no	—
12	C₂×C₇×M₃₂	C₂²	1	2	yes	—
13	C₇×#500	—	2	1	no	—
14	C₇×#500	—	2	1	no	—
15	C₇×#500	C₂²	1	2	yes	—
16	C₇×#500	C₂²	1	2	yes	—
17	C₇×#495	—	2	1	no	—
18	C₇×#495	C₂²	1	2	yes	—
19	C₇×#494	C₂²	1	1	yes	characteristic
20	C₇×#493	C₂²	1	1	yes	characteristic
21	C₇×#492	—	2	1	no	—
22	C₂×C₇×#107	—	2	1	no	—
23	C₂×C₇×#107	C₂²	1	2	yes	—
24	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
25	C₇×#358	C₂²	1	8	yes	—
26	C₇×#357	C₂²	1	8	yes	—
27	C₇×#346	C₂²	1	2	yes	—
28	C₄×C₇×C₁₆	C₂²	1	2	yes	—
29	#33707	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂×C₇×#41	—	2	2	no	—
31	C₂²×C₇×Q₈	—	4	1	no	—
32	C₇×#135	—	2	8	no	—
33	C₇×#135	—	4	4	no	—
34	C₂×C₇×Q₁₆	—	2	1	no	—
35	C₂×C₇×Q₁₆	C₂³	1	2	yes	—
36	C₂×C₇×SD₁₆	—	2	2	no	—
37	C₂×C₇×SD₁₆	—	2	2	no	—
38	C₂×C₇×D₈	—	2	1	no	—
39	C₂×C₇×D₈	C₂³	1	2	yes	—
40	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
41	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
42	C₇×C₄⋊Q₈	—	2	2	no	—
43	C₇×C₄⋊Q₈	—	2	2	no	—
44	C₇×C₄⋊Q₈	—	2	1	no	—
45	C₇×C₄⋊Q₈	—	4	1	no	—
46	C₇×C₄⋊Q₈	—	4	1	no	—
47	C₇×#127	—	2	1	no	—
48	C₇×#124	—	2	2	no	—
49	C₇×#121	—	4	1	no	—
50	C₇×#121	—	4	1	no	—
51	C₄×C₇×D₄	—	4	1	no	—
52	C₄×C₇×D₄	—	4	1	no	—
53	C₂×C₇×C₄⋊C₄	—	2	2	no	—
54	C₂×C₇×C₄⋊C₄	—	4	1	no	—
55	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
56	C₇×M₃₂	—	2	4	no	—
57	C₂×C₇×C₁₆	C₂³	1	4	yes	—
58	C₇×#107	—	2	2	no	—
59	C₇×#107	—	2	2	no	—
60	C₇×#107	—	2	2	no	—
61	C₇×#107	C₂³	1	4	yes	—
62	C₄×C₇×C₈	D₄	1	1	yes	characteristic
63	C₄×C₇×C₈	D₄	1	1	yes	characteristic
64	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
65	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
66	C₂×#500	C₂×C₇	1	1	yes	characteristic, complemented
67	#3147	C₂×C₇	1	1	yes	characteristic
68	#2243	C₂×C₇	1	4	yes	complemented
69	#2241	C₂×C₇	1	2	yes	complemented
70	#2240	C₂×C₇	1	2	yes	complemented
71	#2139	C₂×C₇	1	4	yes	complemented
72	C₄×M₃₂	C₂×C₇	1	1	yes	characteristic, complemented
73	C₇×#41	—	4	4	no	—
74	C₇×#41	—	4	4	no	—
75	C₂×C₇×Q₈	—	2	2	no	—
76	C₂×C₇×Q₈	—	4	2	no	—
77	C₂×C₇×Q₈	—	4	2	no	—
78	C₂×C₇×Q₈	—	8	4	no	—
79	C₂×C₇×D₄	—	2	2	no	—
80	C₂×C₇×D₄	—	4	2	no	—
81	C₂×C₇×D₄	—	4	1	no	—
82	C₂×C₇×D₄	—	4	1	no	—
83	C₂²×C₄×C₇	—	4	2	no	—
84	C₂²×C₄×C₇	—	4	2	no	—
85	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
86	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
87	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
88	C₇×Q₁₆	—	2	4	no	—
89	C₇×Q₁₆	—	4	2	no	—
90	C₇×SD₁₆	—	4	4	no	—
91	C₇×SD₁₆	—	4	4	no	—
92	C₇×D₈	—	2	4	no	—
93	C₇×D₈	—	4	2	no	—
94	C₂×C₇×C₈	SD₁₆	1	4	yes	—
95	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
96	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
98	C₂×C₇×C₈	#41	1	4	yes	—
99	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
100	C₇×C₄⋊C₄	—	2	4	no	—
101	C₇×C₄⋊C₄	—	4	2	no	—
102	C₇×C₄⋊C₄	—	4	2	no	—
103	C₇×C₄⋊C₄	—	4	2	no	—
104	C₇×C₄⋊C₄	—	4	1	no	—
105	C₇×C₄⋊C₄	—	4	1	no	—
106	C₇×C₂²⋊C₄	—	4	2	no	—
107	C₇×C₂²⋊C₄	—	4	2	no	—
108	C₄²×C₇	D₈	1	1	yes	characteristic
109	C₄²×C₇	D₈	1	1	yes	characteristic
110	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
112	C₇×C₁₆	—	2	4	no	—
113	C₂×#135	—	2	1	no	—
114	C₂×#135	C₂²×C₇	1	2	yes	—
115	#532	—	2	1	no	—
116	C₂×C₄⋊Q₈	—	2	1	no	—
117	C₂×M₃₂	C₂²×C₇	1	2	yes	—
118	#500	—	2	1	no	—
119	#500	—	2	1	no	—
120	#500	C₂²×C₇	1	2	yes	—
121	#500	C₂²×C₇	1	2	yes	—
122	#495	—	2	1	no	—
123	#495	C₂²×C₇	1	2	yes	—
124	#494	C₂²×C₇	1	1	yes	characteristic
125	#493	C₂²×C₇	1	1	yes	characteristic
126	#492	—	2	1	no	—
127	C₂×#107	—	2	1	no	—
128	C₂×#107	C₂²×C₇	1	2	yes	—
129	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
130	#358	C₂²×C₇	1	8	yes	—
131	#357	C₂²×C₇	1	8	yes	—
132	#346	C₂²×C₇	1	2	yes	—
133	C₄×C₁₆	C₂²×C₇	1	2	yes	—
134	C₂³×C₇	—	4	2	no	—
135	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
136	C₇×Q₈	—	4	2	no	—
137	C₇×Q₈	—	4	2	no	—
138	C₇×Q₈	—	8	4	no	—
139	C₇×Q₈	—	8	4	no	—
140	C₇×D₄	—	4	2	no	—
141	C₇×D₄	—	4	2	no	—
142	C₇×D₄	—	8	4	no	—
143	C₇×D₄	—	8	2	no	—
144	C₇×D₄	—	8	2	no	—
145	C₂×C₄×C₇	—	2	2	no	—
146	C₂×C₄×C₇	—	2	2	no	—
147	C₂×C₄×C₇	—	4	2	no	—
148	C₂×C₄×C₇	—	4	4	no	—
149	C₂×C₄×C₇	—	8	4	no	—
150	C₂×C₄×C₇	—	8	4	no	—
151	C₂×C₄×C₇	#124	1	2	yes	—
152	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
153	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
154	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
156	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
157	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
158	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
159	C₇×C₈	#124	1	2	yes	—
160	C₇×C₈	C₂×SD₁₆	1	2	yes	—
161	C₇×C₈	C₂×SD₁₆	1	2	yes	—
162	C₇×C₈	C₂×#41	1	2	yes	—
163	C₂×#41	—	2	2	no	—
164	C₂²×Q₈	—	4	1	no	—
165	#135	—	2	8	no	—
166	#135	—	4	4	no	—
167	C₂×Q₁₆	—	2	1	no	—
168	C₂×Q₁₆	C₂³×C₇	1	2	yes	—
169	C₂×SD₁₆	—	2	2	no	—
170	C₂×SD₁₆	—	2	2	no	—
171	C₂×D₈	—	2	1	no	—
172	C₂×D₈	C₂³×C₇	1	2	yes	—
173	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
174	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
175	C₄⋊Q₈	—	2	2	no	—
176	C₄⋊Q₈	—	2	2	no	—
177	C₄⋊Q₈	—	2	1	no	—
178	C₄⋊Q₈	—	4	1	no	—
179	C₄⋊Q₈	—	4	1	no	—
180	#127	—	2	1	no	—
181	#124	—	2	2	no	—
182	#121	—	4	1	no	—
183	#121	—	4	1	no	—
184	C₄×D₄	—	4	1	no	—
185	C₄×D₄	—	4	1	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	—	4	1	no	—
188	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
189	M₃₂	—	2	4	no	—
190	C₂×C₁₆	C₂³×C₇	1	4	yes	—
191	#107	—	2	2	no	—
192	#107	—	2	2	no	—
193	#107	—	2	2	no	—
194	#107	C₂³×C₇	1	4	yes	—
195	C₄×C₈	C₇×D₄	1	1	yes	characteristic
196	C₄×C₈	C₇×D₄	1	1	yes	characteristic
197	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
198	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
199	C₂²×C₇	—	2	1	no	—
200	C₂²×C₇	—	2	1	no	—
201	C₂²×C₇	—	8	2	no	—
202	C₂²×C₇	—	8	2	no	—
203	C₂²×C₇	—	8	2	no	—
204	C₂²×C₇	#486	1	2	yes	—
205	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
206	C₄×C₇	—	2	1	no	—
207	C₄×C₇	—	2	1	no	—
208	C₄×C₇	—	8	2	no	—
209	C₄×C₇	—	8	4	no	—
210	C₄×C₇	#486	1	2	yes	—
211	C₄×C₇	#509	1	2	yes	—
212	C₄×C₇	#510	1	2	yes	—
213	C₄×C₇	C₂×#124	1	1	yes	characteristic
214	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
215	#41	—	4	4	no	—
216	#41	—	4	4	no	—
217	C₂×Q₈	—	2	2	no	—
218	C₂×Q₈	—	4	2	no	—
219	C₂×Q₈	—	4	2	no	—
220	C₂×Q₈	—	8	4	no	—
221	C₂×D₄	—	2	2	no	—
222	C₂×D₄	—	4	2	no	—
223	C₂×D₄	—	4	1	no	—
224	C₂×D₄	—	4	1	no	—
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	—	4	2	no	—
227	C₂²×C₄	C₇×D₈	1	1	yes	characteristic
228	C₂²×C₄	C₇×D₈	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
230	Q₁₆	—	2	4	no	—
231	Q₁₆	—	4	2	no	—
232	SD₁₆	—	4	4	no	—
233	SD₁₆	—	4	4	no	—
234	D₈	—	2	4	no	—
235	D₈	—	4	2	no	—
236	C₂×C₈	C₇×SD₁₆	1	4	yes	—
237	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
238	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
239	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
240	C₂×C₈	C₇×#41	1	4	yes	—
241	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	—	4	1	no	—
247	C₄⋊C₄	—	4	1	no	—
248	C₂²⋊C₄	—	4	2	no	—
249	C₂²⋊C₄	—	4	2	no	—
250	C₄²	C₇×D₈	1	1	yes	characteristic
251	C₄²	C₇×D₈	1	1	yes	characteristic
252	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
253	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
254	C₁₆	—	2	4	no	—
255	C₂×C₇	—	2	2	no	—
256	C₂×C₇	—	16	2	no	—
257	C₂×C₇	C₂×#486	1	1	yes	characteristic
258	C₂×C₇	C₂×#509	1	1	yes	characteristic
259	C₂×C₇	C₂×#510	1	1	yes	characteristic
260	C₂³	—	4	2	no	—
261	C₂³	C₂×C₇×D₈	1	1	yes	characteristic
262	Q₈	—	4	2	no	—
263	Q₈	—	4	2	no	—
264	Q₈	—	8	4	no	—
265	Q₈	—	8	4	no	—
266	D₄	—	4	2	no	—
267	D₄	—	4	2	no	—
268	D₄	—	8	4	no	—
269	D₄	—	8	2	no	—
270	D₄	—	8	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	8	4	no	—
276	C₂×C₄	—	8	4	no	—
277	C₂×C₄	C₇×#124	1	2	yes	—
278	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
279	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
280	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
281	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
282	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
283	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
284	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
285	C₈	C₇×#124	1	2	yes	—
286	C₈	C₂×C₇×SD₁₆	1	2	yes	—
287	C₈	C₂×C₇×SD₁₆	1	2	yes	—
288	C₈	C₂×C₇×#41	1	2	yes	—
289	C₇	#33707	1	1	yes	characteristic, direct product factor, Sylow
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	8	2	no	—
293	C₂²	—	8	2	no	—
294	C₂²	—	8	2	no	—
295	C₂²	C₇×#486	1	2	yes	—
296	C₂²	C₂²×C₇×D₈	1	1	yes	characteristic
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	8	4	no	—
300	C₄	—	8	2	no	—
301	C₄	C₇×#486	1	2	yes	—
302	C₄	C₇×#509	1	2	yes	—
303	C₄	C₇×#510	1	2	yes	—
304	C₄	C₂×C₇×#124	1	1	yes	characteristic
305	C₄	C₂²×C₇×SD₁₆	1	1	yes	characteristic
306	C₂	—	2	2	no	—
307	C₂	—	16	2	no	—
308	C₂	C₂×C₇×#486	1	1	yes	characteristic
309	C₂	C₂×C₇×#509	1	1	yes	characteristic
310	C₂	C₂×C₇×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₇
prime radical	C₇×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×#500	C₂	semidirect
3. C₇×#2243	C₂	semidirect
4. C₇×#2241	C₂	semidirect
5. C₇×#2240	C₂	semidirect
6. C₇×#2139	C₂	semidirect
7. C₄×C₇×M₃₂	C₂	semidirect
29. #33707	C₇	direct (N × H)
66. C₂×#500	C₂×C₇	semidirect
68. #2243	C₂×C₇	semidirect
69. #2241	C₂×C₇	semidirect
70. #2240	C₂×C₇	semidirect
71. #2139	C₂×C₇	semidirect
72. C₄×M₃₂	C₂×C₇	semidirect
289. C₇	#33707	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×#500	2
C₇×#3147	2
C₇×#2243	2
C₇×#2241	2
C₇×#2240	2
C₇×#2139	2
C₄×C₇×M₃₂	2
#33707	7

Subgroups — C7×#33707 #13210923

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#33707 #13210923