Subgroups — C₇×#5832 #13061547

Order 1344.

620 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₇×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#235	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#947	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#944	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#940	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#939	C₂	1	4	yes	maximal, complemented
7	C₇×#936	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₇×#67	C₂	1	4	yes	maximal, complemented
9	C₂×C₄×C₇×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#532	—	3	1	no	maximal, Hall
11	C₃×C₇×#41	—	2	4	no	—
12	C₂×C₃×C₇×Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₇×D₄	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₇×D₄	C₂²	1	2	yes	—
15	C₂²×C₃×C₄×C₇	C₂²	1	2	yes	—
16	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
17	C₂×C₇×#67	C₂²	1	4	yes	—
18	C₂×C₇×#67	C₂²	1	2	yes	—
19	C₂²×C₇×Dic₃	C₂²	1	2	yes	—
20	C₇×#235	—	2	4	no	—
21	C₂×C₇×D₁₂	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₃×C₇	C₂²	1	2	yes	—
23	C₂×C₇×Dic₆	C₂²	1	1	yes	characteristic
24	C₇×#217	C₂²	1	4	yes	—
25	C₇×(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
26	C₇×Dic₃⋊C₄	C₂²	1	4	yes	—
27	C₄×C₇×Dic₃	C₂²	1	2	yes	—
28	C₄×C₇×Dic₃	C₂²	1	1	yes	characteristic
29	C₄×C₇×Dic₃	C₂²	1	1	yes	characteristic
30	C₂×C₇×#41	—	3	1	no	—
31	C₂×C₇×#41	D₃	1	1	yes	characteristic, complemented
32	C₇×C₄⋊Q₈	—	3	1	no	—
33	C₇×#127	—	3	1	no	—
34	C₇×#124	—	3	1	no	—
35	C₇×#124	—	3	1	no	—
36	C₇×#121	—	3	4	no	—
37	C₄×C₇×D₄	—	3	4	no	—
38	C₂×C₄²×C₇	—	3	1	no	—
39	#5832	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
40	C₂³×C₃×C₇	C₂³	1	2	yes	—
41	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
42	C₂²×D₃×C₇	C₂³	1	2	yes	—
43	C₃×C₇×Q₈	—	2	2	no	—
44	C₃×C₇×D₄	—	2	4	no	—
45	C₃×C₇×D₄	—	2	2	no	—
46	C₂×C₃×C₄×C₇	—	2	4	no	—
47	C₂×C₃×C₄×C₇	D₄	1	4	yes	complemented
48	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
49	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
50	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
51	C₇×#67	—	2	4	no	—
52	C₇×#67	—	2	8	no	—
53	C₂×C₇×Dic₃	—	2	4	no	—
54	C₂×C₇×Dic₃	C₂³	1	2	yes	—
55	C₂×C₇×Dic₃	C₂³	1	2	yes	—
56	C₂×C₇×Dic₃	C₂³	1	2	yes	—
57	C₇×D₁₂	—	2	2	no	—
58	C₄×D₃×C₇	—	2	4	no	—
59	C₇×Dic₆	—	2	2	no	—
60	C₇×#41	—	2	4	no	—
61	C₇×#41	—	6	4	no	—
62	C₂×C₇×Q₈	—	3	1	no	—
63	C₂×C₇×Q₈	C₂×D₃	1	1	yes	characteristic
64	C₂×C₇×D₄	—	3	4	no	—
65	C₂×C₇×D₄	—	3	2	no	—
66	C₂×C₇×D₄	—	3	1	no	—
67	C₂×C₇×D₄	C₂×D₃	1	2	yes	—
68	C₂×C₇×D₄	C₂×D₃	1	1	yes	characteristic
69	C₂²×C₄×C₇	—	3	2	no	—
70	C₂²×C₄×C₇	—	3	2	no	—
71	C₂²×C₄×C₇	C₂×D₃	1	2	yes	—
72	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
73	C₇×C₄⋊C₄	—	3	4	no	—
74	C₇×C₂²⋊C₄	—	3	4	no	—
75	C₇×C₂²⋊C₄	—	3	4	no	—
76	C₄²×C₇	—	3	1	no	—
77	C₄²×C₇	—	3	1	no	—
78	C₄²×C₇	—	3	2	no	—
79	C₂×C₃×#41	C₂×C₇	1	1	yes	characteristic, complemented
80	C₂×#235	C₂×C₇	1	1	yes	characteristic, complemented
81	#947	C₂×C₇	1	1	yes	characteristic, complemented
82	#944	C₂×C₇	1	1	yes	characteristic, complemented
83	#940	C₂×C₇	1	1	yes	characteristic, complemented
84	#939	C₂×C₇	1	4	yes	complemented
85	#936	C₂×C₇	1	1	yes	characteristic, complemented
86	C₄×#67	C₂×C₇	1	4	yes	complemented
87	C₂×C₄×Dic₃	C₂×C₇	1	1	yes	characteristic, complemented
88	C₂²×C₃×C₇	—	2	2	no	—
89	C₂²×C₃×C₇	—	2	2	no	—
90	C₂²×C₃×C₇	—	2	4	no	—
91	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
92	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
93	C₂×D₃×C₇	—	2	2	no	—
94	C₂×D₃×C₇	—	2	4	no	—
95	C₃×C₄×C₇	—	2	2	no	—
96	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
97	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
98	C₇×Dic₃	—	2	2	no	—
99	C₇×Dic₃	—	2	2	no	—
100	C₇×Dic₃	#41	1	4	yes	complemented
101	#532	—	3	1	no	Sylow
102	C₂³×C₇	—	3	2	no	—
103	C₂³×C₇	C₂²×D₃	1	2	yes	—
104	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
105	C₇×Q₈	—	2	2	no	—
106	C₇×Q₈	—	6	2	no	—
107	C₇×D₄	—	2	4	no	—
108	C₇×D₄	—	2	2	no	—
109	C₇×D₄	—	6	8	no	—
110	C₇×D₄	—	6	4	no	—
111	C₇×D₄	—	6	2	no	—
112	C₂×C₄×C₇	—	2	4	no	—
113	C₂×C₄×C₇	—	3	2	no	—
114	C₂×C₄×C₇	—	3	2	no	—
115	C₂×C₄×C₇	—	3	2	no	—
116	C₂×C₄×C₇	—	6	4	no	—
117	C₂×C₄×C₇	—	6	4	no	—
118	C₂×C₄×C₇	#67	1	4	yes	complemented
119	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
120	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
121	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
122	C₃×#41	—	2	4	no	—
123	C₂×C₃×Q₈	C₂²×C₇	1	1	yes	characteristic
124	C₂×C₃×D₄	C₂²×C₇	1	1	yes	characteristic
125	C₂×C₃×D₄	C₂²×C₇	1	2	yes	—
126	C₂²×C₃×C₄	C₂²×C₇	1	2	yes	—
127	C₂²×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
128	C₂×#67	C₂²×C₇	1	4	yes	—
129	C₂×#67	C₂²×C₇	1	2	yes	—
130	C₂²×Dic₃	C₂²×C₇	1	2	yes	—
131	#235	—	2	4	no	—
132	C₂×D₁₂	C₂²×C₇	1	1	yes	characteristic
133	C₂×C₄×D₃	C₂²×C₇	1	2	yes	—
134	C₂×Dic₆	C₂²×C₇	1	1	yes	characteristic
135	#217	C₂²×C₇	1	4	yes	—
136	(C₂×D₃)⋊C₄	C₂²×C₇	1	4	yes	—
137	Dic₃⋊C₄	C₂²×C₇	1	4	yes	—
138	C₄×Dic₃	C₂²×C₇	1	2	yes	—
139	C₄×Dic₃	C₂²×C₇	1	1	yes	characteristic
140	C₄×Dic₃	C₂²×C₇	1	1	yes	characteristic
141	C₂×C₃×C₇	—	2	2	no	—
142	C₂×C₃×C₇	—	4	2	no	—
143	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₃×C₇	C₂×#41	1	2	yes	—
145	D₃×C₇	—	4	2	no	—
146	C₂×#41	—	3	1	no	—
147	C₂×#41	D₃×C₇	1	1	yes	characteristic, complemented
148	C₄⋊Q₈	—	3	1	no	—
149	#127	—	3	1	no	—
150	#124	—	3	1	no	—
151	#124	—	3	1	no	—
152	#121	—	3	4	no	—
153	C₄×D₄	—	3	4	no	—
154	C₂×C₄²	—	3	1	no	—
155	C₂²×C₇	—	2	2	no	—
156	C₂²×C₇	—	2	4	no	—
157	C₂²×C₇	—	2	2	no	—
158	C₂²×C₇	—	6	4	no	—
159	C₂²×C₇	—	6	2	no	—
160	C₂²×C₇	C₂×#67	1	2	yes	—
161	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
162	C₄×C₇	—	2	2	no	—
163	C₄×C₇	—	3	4	no	—
164	C₄×C₇	—	6	2	no	—
165	C₄×C₇	—	6	2	no	—
166	C₄×C₇	C₂×#67	1	2	yes	—
167	C₄×C₇	C₂×#67	1	2	yes	—
168	C₂³×C₃	C₂³×C₇	1	2	yes	—
169	C₂³×C₃	C₂³×C₇	1	1	yes	characteristic
170	C₂²×D₃	C₂³×C₇	1	2	yes	—
171	C₃×Q₈	—	2	2	no	—
172	C₃×D₄	—	2	4	no	—
173	C₃×D₄	—	2	2	no	—
174	C₂×C₃×C₄	—	2	4	no	—
175	C₂×C₃×C₄	C₇×D₄	1	4	yes	complemented
176	C₂×C₃×C₄	C₂³×C₇	1	2	yes	—
177	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
179	#67	—	2	4	no	—
180	#67	—	2	8	no	—
181	C₂×Dic₃	—	2	4	no	—
182	C₂×Dic₃	C₂³×C₇	1	2	yes	—
183	C₂×Dic₃	C₂³×C₇	1	2	yes	—
184	C₂×Dic₃	C₂³×C₇	1	2	yes	—
185	D₁₂	—	2	2	no	—
186	C₄×D₃	—	2	4	no	—
187	Dic₆	—	2	2	no	—
188	C₃×C₇	#532	1	1	yes	characteristic, complemented, Hall
189	#41	—	2	4	no	—
190	#41	—	6	4	no	—
191	C₂×Q₈	—	3	1	no	—
192	C₂×Q₈	C₂×D₃×C₇	1	1	yes	characteristic
193	C₂×D₄	—	3	4	no	—
194	C₂×D₄	—	3	2	no	—
195	C₂×D₄	—	3	1	no	—
196	C₂×D₄	C₂×D₃×C₇	1	2	yes	—
197	C₂×D₄	C₂×D₃×C₇	1	1	yes	characteristic
198	C₂²×C₄	—	3	2	no	—
199	C₂²×C₄	—	3	2	no	—
200	C₂²×C₄	C₂×D₃×C₇	1	2	yes	—
201	C₂²×C₄	C₂×D₃×C₇	1	1	yes	characteristic
202	C₄⋊C₄	—	3	4	no	—
203	C₂²⋊C₄	—	3	4	no	—
204	C₂²⋊C₄	—	3	4	no	—
205	C₄²	—	3	2	no	—
206	C₄²	—	3	1	no	—
207	C₄²	—	3	1	no	—
208	C₂×C₇	—	2	2	no	—
209	C₂×C₇	—	4	2	no	—
210	C₂×C₇	—	12	2	no	—
211	C₂×C₇	D₃×#41	1	2	yes	—
212	C₂×C₇	C₂²×#67	1	1	yes	characteristic
213	C₂²×C₃	—	2	2	no	—
214	C₂²×C₃	—	2	2	no	—
215	C₂²×C₃	—	2	4	no	—
216	C₂²×C₃	C₂×C₇×D₄	1	2	yes	—
217	C₂²×C₃	C₂⁴×C₇	1	1	yes	characteristic
218	C₂×D₃	—	2	2	no	—
219	C₂×D₃	—	2	4	no	—
220	C₃×C₄	—	2	2	no	—
221	C₃×C₄	C₂×C₇×D₄	1	2	yes	—
222	C₃×C₄	C₂×C₇×D₄	1	2	yes	—
223	Dic₃	—	2	2	no	—
224	Dic₃	—	2	2	no	—
225	Dic₃	C₇×#41	1	4	yes	complemented
226	C₂³	—	3	2	no	—
227	C₂³	C₂²×D₃×C₇	1	2	yes	—
228	C₂³	C₂²×D₃×C₇	1	1	yes	characteristic
229	Q₈	—	2	2	no	—
230	Q₈	—	6	2	no	—
231	D₄	—	2	4	no	—
232	D₄	—	2	2	no	—
233	D₄	—	6	8	no	—
234	D₄	—	6	4	no	—
235	D₄	—	6	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	3	2	no	—
238	C₂×C₄	—	3	2	no	—
239	C₂×C₄	—	3	2	no	—
240	C₂×C₄	—	6	4	no	—
241	C₂×C₄	—	6	4	no	—
242	C₂×C₄	C₇×#67	1	4	yes	complemented
243	C₂×C₄	C₂²×D₃×C₇	1	2	yes	—
244	C₂×C₄	C₂²×D₃×C₇	1	1	yes	characteristic
245	C₂×C₄	C₂²×D₃×C₇	1	1	yes	characteristic
246	C₇	#5832	1	1	yes	characteristic, direct product factor, Sylow
247	C₂×C₃	—	2	2	no	—
248	C₂×C₃	—	4	2	no	—
249	C₂×C₃	C₂²×C₇×D₄	1	1	yes	characteristic
250	C₂×C₃	C₂×C₇×#41	1	2	yes	—
251	D₃	—	4	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	6	4	no	—
256	C₂²	—	6	2	no	—
257	C₂²	C₂×C₇×#67	1	2	yes	—
258	C₂²	C₂³×D₃×C₇	1	1	yes	characteristic
259	C₄	—	2	2	no	—
260	C₄	—	3	4	no	—
261	C₄	—	6	2	no	—
262	C₄	—	6	2	no	—
263	C₄	C₂×C₇×#67	1	2	yes	—
264	C₄	C₂×C₇×#67	1	2	yes	—
265	C₃	C₇×#532	1	1	yes	characteristic, complemented, Sylow
266	C₂	—	2	2	no	—
267	C₂	—	4	2	no	—
268	C₂	—	12	2	no	—
269	C₂	D₃×C₇×#41	1	2	yes	—
270	C₂	C₂²×C₇×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₇
Frattini	C₂²	C₂³×D₃×C₇
Fitting	C₂×C₃×C₇×#41	C₂
socle	C₂²×C₃×C₇	C₂⁴
hypercenter	C₂×C₇×#41	D₃
hypocenter	C₃	C₇×#532
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₇×#41

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₇×#41	C₂	semidirect
2. C₂×C₇×#235	C₂	semidirect
3. C₇×#947	C₂	semidirect
4. C₇×#944	C₂	semidirect
5. C₇×#940	C₂	semidirect
6. C₇×#939	C₂	semidirect
7. C₇×#936	C₂	semidirect
8. C₄×C₇×#67	C₂	semidirect
9. C₂×C₄×C₇×Dic₃	C₂	semidirect
31. C₂×C₇×#41	D₃	semidirect
39. #5832	C₇	direct (N × H)
47. C₂×C₃×C₄×C₇	D₄	semidirect
79. C₂×C₃×#41	C₂×C₇	semidirect
80. C₂×#235	C₂×C₇	semidirect
81. #947	C₂×C₇	semidirect
82. #944	C₂×C₇	semidirect
83. #940	C₂×C₇	semidirect
84. #939	C₂×C₇	semidirect
85. #936	C₂×C₇	semidirect
86. C₄×#67	C₂×C₇	semidirect
87. C₂×C₄×Dic₃	C₂×C₇	semidirect
100. C₇×Dic₃	#41	semidirect
118. C₂×C₄×C₇	#67	semidirect
147. C₂×#41	D₃×C₇	semidirect
175. C₂×C₃×C₄	C₇×D₄	semidirect
188. C₃×C₇	#532	semidirect
225. Dic₃	C₇×#41	semidirect
242. C₂×C₄	C₇×#67	semidirect
246. C₇	#5832	direct (N × H)
265. C₃	C₇×#532	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₇×#41	2
C₂×C₇×#235	2
C₇×#947	2
C₇×#944	2
C₇×#940	2
C₇×#939	2
C₇×#936	2
C₄×C₇×#67	2
C₂×C₄×C₇×Dic₃	2
C₇×#532	3
#5832	7

Subgroups — C7×#5832 #13061547

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#5832 #13061547