Subgroups — C₇×#5833 #13061548

Order 1344.

588 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×#946	C₂	1	2	yes	maximal, complemented
4	C₇×#939	C₂	1	4	yes	maximal, complemented
5	C₇×#938	C₂	1	2	yes	maximal, complemented
6	C₇×#930	C₂	1	2	yes	maximal, complemented
7	C₂×C₇×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#924	C₂	1	2	yes	maximal, complemented
9	C₇×#537	—	3	1	no	maximal, Hall
10	C₃×C₇×#41	—	2	4	no	—
11	C₂×C₃×C₇×Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₇×D₄	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₇×D₄	C₂²	1	2	yes	—
14	C₂²×C₃×C₄×C₇	C₂²	1	2	yes	—
15	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
16	C₂×C₇×#67	C₂²	1	2	yes	—
17	C₂×C₇×#67	C₂²	1	4	yes	—
18	C₂²×C₇×Dic₃	C₂²	1	2	yes	—
19	C₇×#235	—	2	4	no	—
20	C₂×C₇×D₁₂	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₃×C₇	C₂²	1	2	yes	—
22	C₂×C₇×Dic₆	C₂²	1	1	yes	characteristic
23	C₇×#217	C₂²	1	4	yes	—
24	C₇×(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
25	C₇×C₄⋊Dic₃	C₂²	1	2	yes	—
26	C₇×C₄⋊Dic₃	C₂²	1	2	yes	—
27	C₇×Dic₃⋊C₄	C₂²	1	4	yes	—
28	C₂×C₇×#41	—	3	1	no	—
29	C₂×C₇×#41	D₃	1	1	yes	characteristic, complemented
30	C₇×C₂²⋊Q₈	—	3	2	no	—
31	C₇×C₂²⋊Q₈	—	3	2	no	—
32	C₇×#121	—	3	2	no	—
33	C₇×#121	—	3	4	no	—
34	C₇×#121	—	3	2	no	—
35	C₂×C₇×C₄⋊C₄	—	3	1	no	—
36	#5833	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
37	C₂³×C₃×C₇	C₂³	1	2	yes	—
38	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
39	C₂²×D₃×C₇	C₂³	1	2	yes	—
40	C₃×C₇×Q₈	—	2	2	no	—
41	C₃×C₇×D₄	—	2	4	no	—
42	C₃×C₇×D₄	—	2	2	no	—
43	C₂×C₃×C₄×C₇	—	2	4	no	—
44	C₂×C₃×C₄×C₇	D₄	1	4	yes	complemented
45	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
46	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
47	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
48	C₇×#67	—	2	4	no	—
49	C₇×#67	—	4	4	no	—
50	C₂×C₇×Dic₃	—	4	2	no	—
51	C₂×C₇×Dic₃	C₂³	1	2	yes	—
52	C₂×C₇×Dic₃	C₂³	1	4	yes	—
53	C₇×D₁₂	—	2	2	no	—
54	C₄×D₃×C₇	—	2	4	no	—
55	C₇×Dic₆	—	2	2	no	—
56	C₇×#41	—	2	4	no	—
57	C₇×#41	—	6	4	no	—
58	C₂×C₇×Q₈	—	3	1	no	—
59	C₂×C₇×Q₈	C₂×D₃	1	1	yes	characteristic
60	C₂×C₇×D₄	—	3	2	no	—
61	C₂×C₇×D₄	—	3	4	no	—
62	C₂×C₇×D₄	—	3	1	no	—
63	C₂×C₇×D₄	C₂×D₃	1	2	yes	—
64	C₂×C₇×D₄	C₂×D₃	1	1	yes	characteristic
65	C₂²×C₄×C₇	—	3	2	no	—
66	C₂²×C₄×C₇	—	3	2	no	—
67	C₂²×C₄×C₇	C₂×D₃	1	2	yes	—
68	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
69	C₇×C₄⋊C₄	—	3	4	no	—
70	C₇×C₄⋊C₄	—	3	2	no	—
71	C₇×C₄⋊C₄	—	3	2	no	—
72	C₇×C₂²⋊C₄	—	3	4	no	—
73	C₇×C₂²⋊C₄	—	3	4	no	—
74	C₂×C₃×#41	C₂×C₇	1	1	yes	characteristic, complemented
75	C₂×#235	C₂×C₇	1	1	yes	characteristic, complemented
76	#946	C₂×C₇	1	2	yes	complemented
77	#939	C₂×C₇	1	4	yes	complemented
78	#938	C₂×C₇	1	2	yes	complemented
79	#930	C₂×C₇	1	2	yes	complemented
80	C₂×C₄⋊Dic₃	C₂×C₇	1	1	yes	characteristic, complemented
81	#924	C₂×C₇	1	2	yes	complemented
82	C₂²×C₃×C₇	—	2	1	no	—
83	C₂²×C₃×C₇	—	2	1	no	—
84	C₂²×C₃×C₇	—	2	2	no	—
85	C₂²×C₃×C₇	—	2	2	no	—
86	C₂²×C₃×C₇	—	2	2	no	—
87	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
88	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
89	C₂×D₃×C₇	—	2	2	no	—
90	C₂×D₃×C₇	—	2	2	no	—
91	C₂×D₃×C₇	—	2	2	no	—
92	C₃×C₄×C₇	—	2	2	no	—
93	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
94	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
95	C₇×Dic₃	—	2	2	no	—
96	C₇×Dic₃	—	2	4	no	—
97	#537	—	3	1	no	Sylow
98	C₂³×C₇	—	3	2	no	—
99	C₂³×C₇	C₂²×D₃	1	2	yes	—
100	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
101	C₇×Q₈	—	2	2	no	—
102	C₇×Q₈	—	6	2	no	—
103	C₇×D₄	—	2	4	no	—
104	C₇×D₄	—	2	2	no	—
105	C₇×D₄	—	6	4	no	—
106	C₇×D₄	—	6	2	no	—
107	C₇×D₄	—	12	4	no	—
108	C₂×C₄×C₇	—	2	4	no	—
109	C₂×C₄×C₇	—	3	4	no	—
110	C₂×C₄×C₇	—	3	2	no	—
111	C₂×C₄×C₇	—	6	4	no	—
112	C₂×C₄×C₇	—	12	2	no	—
113	C₂×C₄×C₇	#67	1	4	yes	complemented
114	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
115	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
116	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
117	C₃×#41	—	2	4	no	—
118	C₂×C₃×Q₈	C₂²×C₇	1	1	yes	characteristic
119	C₂×C₃×D₄	C₂²×C₇	1	1	yes	characteristic
120	C₂×C₃×D₄	C₂²×C₇	1	2	yes	—
121	C₂²×C₃×C₄	C₂²×C₇	1	2	yes	—
122	C₂²×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
123	C₂×#67	C₂²×C₇	1	2	yes	—
124	C₂×#67	C₂²×C₇	1	4	yes	—
125	C₂²×Dic₃	C₂²×C₇	1	2	yes	—
126	#235	—	2	4	no	—
127	C₂×D₁₂	C₂²×C₇	1	1	yes	characteristic
128	C₂×C₄×D₃	C₂²×C₇	1	2	yes	—
129	C₂×Dic₆	C₂²×C₇	1	1	yes	characteristic
130	#217	C₂²×C₇	1	4	yes	—
131	(C₂×D₃)⋊C₄	C₂²×C₇	1	4	yes	—
132	C₄⋊Dic₃	C₂²×C₇	1	2	yes	—
133	C₄⋊Dic₃	C₂²×C₇	1	2	yes	—
134	Dic₃⋊C₄	C₂²×C₇	1	4	yes	—
135	C₂×C₃×C₇	—	2	2	no	—
136	C₂×C₃×C₇	—	4	2	no	—
137	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₃×C₇	H(5,𝔽₂)	1	1	yes	characteristic
139	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
140	D₃×C₇	—	4	2	no	—
141	C₂×#41	—	3	1	no	—
142	C₂×#41	D₃×C₇	1	1	yes	characteristic, complemented
143	C₂²⋊Q₈	—	3	2	no	—
144	C₂²⋊Q₈	—	3	2	no	—
145	#121	—	3	2	no	—
146	#121	—	3	4	no	—
147	#121	—	3	2	no	—
148	C₂×C₄⋊C₄	—	3	1	no	—
149	C₂²×C₇	—	2	1	no	—
150	C₂²×C₇	—	2	1	no	—
151	C₂²×C₇	—	2	2	no	—
152	C₂²×C₇	—	2	2	no	—
153	C₂²×C₇	—	2	2	no	—
154	C₂²×C₇	—	6	2	no	—
155	C₂²×C₇	—	6	2	no	—
156	C₂²×C₇	—	6	2	no	—
157	C₂²×C₇	C₂×#67	1	2	yes	—
158	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
159	C₄×C₇	—	2	2	no	—
160	C₄×C₇	—	6	4	no	—
161	C₄×C₇	—	6	2	no	—
162	C₄×C₇	C₂×#67	1	2	yes	—
163	C₄×C₇	C₂×#67	1	2	yes	—
164	C₂³×C₃	C₂³×C₇	1	2	yes	—
165	C₂³×C₃	C₂³×C₇	1	1	yes	characteristic
166	C₂²×D₃	C₂³×C₇	1	2	yes	—
167	C₃×Q₈	—	2	2	no	—
168	C₃×D₄	—	2	4	no	—
169	C₃×D₄	—	2	2	no	—
170	C₂×C₃×C₄	—	2	4	no	—
171	C₂×C₃×C₄	C₇×D₄	1	4	yes	complemented
172	C₂×C₃×C₄	C₂³×C₇	1	2	yes	—
173	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
175	#67	—	2	4	no	—
176	#67	—	4	4	no	—
177	C₂×Dic₃	—	4	2	no	—
178	C₂×Dic₃	C₂³×C₇	1	2	yes	—
179	C₂×Dic₃	C₂³×C₇	1	4	yes	—
180	D₁₂	—	2	2	no	—
181	C₄×D₃	—	2	4	no	—
182	Dic₆	—	2	2	no	—
183	C₃×C₇	#537	1	1	yes	characteristic, complemented, Hall
184	#41	—	2	4	no	—
185	#41	—	6	4	no	—
186	C₂×Q₈	—	3	1	no	—
187	C₂×Q₈	C₂×D₃×C₇	1	1	yes	characteristic
188	C₂×D₄	—	3	2	no	—
189	C₂×D₄	—	3	4	no	—
190	C₂×D₄	—	3	1	no	—
191	C₂×D₄	C₂×D₃×C₇	1	2	yes	—
192	C₂×D₄	C₂×D₃×C₇	1	1	yes	characteristic
193	C₂²×C₄	—	3	2	no	—
194	C₂²×C₄	—	3	2	no	—
195	C₂²×C₄	C₂×D₃×C₇	1	2	yes	—
196	C₂²×C₄	C₂×D₃×C₇	1	1	yes	characteristic
197	C₄⋊C₄	—	3	2	no	—
198	C₄⋊C₄	—	3	2	no	—
199	C₄⋊C₄	—	3	4	no	—
200	C₂²⋊C₄	—	3	4	no	—
201	C₂²⋊C₄	—	3	4	no	—
202	C₂×C₇	—	2	2	no	—
203	C₂×C₇	—	4	2	no	—
204	C₂×C₇	—	12	2	no	—
205	C₂×C₇	#1009	1	1	yes	characteristic
206	C₂×C₇	#1010	1	1	yes	characteristic
207	C₂×C₇	C₂²×#67	1	1	yes	characteristic
208	C₂²×C₃	—	2	1	no	—
209	C₂²×C₃	—	2	1	no	—
210	C₂²×C₃	—	2	2	no	—
211	C₂²×C₃	—	2	2	no	—
212	C₂²×C₃	—	2	2	no	—
213	C₂²×C₃	C₂×C₇×D₄	1	2	yes	—
214	C₂²×C₃	C₂⁴×C₇	1	1	yes	characteristic
215	C₂×D₃	—	2	2	no	—
216	C₂×D₃	—	2	2	no	—
217	C₂×D₃	—	2	2	no	—
218	C₃×C₄	—	2	2	no	—
219	C₃×C₄	C₂×C₇×D₄	1	2	yes	—
220	C₃×C₄	C₂×C₇×D₄	1	2	yes	—
221	Dic₃	—	2	2	no	—
222	Dic₃	—	2	4	no	—
223	C₂³	—	3	2	no	—
224	C₂³	C₂²×D₃×C₇	1	2	yes	—
225	C₂³	C₂²×D₃×C₇	1	1	yes	characteristic
226	Q₈	—	2	2	no	—
227	Q₈	—	6	2	no	—
228	D₄	—	2	4	no	—
229	D₄	—	2	2	no	—
230	D₄	—	6	4	no	—
231	D₄	—	6	2	no	—
232	D₄	—	12	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	3	4	no	—
235	C₂×C₄	—	3	2	no	—
236	C₂×C₄	—	6	4	no	—
237	C₂×C₄	—	12	2	no	—
238	C₂×C₄	C₇×#67	1	4	yes	complemented
239	C₂×C₄	C₂²×D₃×C₇	1	2	yes	—
240	C₂×C₄	C₂²×D₃×C₇	1	1	yes	characteristic
241	C₂×C₄	C₂²×D₃×C₇	1	1	yes	characteristic
242	C₇	#5833	1	1	yes	characteristic, direct product factor, Sylow
243	C₂×C₃	—	2	2	no	—
244	C₂×C₃	—	4	2	no	—
245	C₂×C₃	C₂²×C₇×D₄	1	1	yes	characteristic
246	C₂×C₃	C₇×H(5,𝔽₂)	1	1	yes	characteristic
247	C₂×C₃	C₇×2₋¹⁺⁴	1	1	yes	characteristic
248	D₃	—	4	2	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	6	2	no	—
255	C₂²	—	6	2	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	—	6	4	no	—
261	C₄	—	6	2	no	—
262	C₄	C₂×C₇×#67	1	2	yes	—
263	C₄	C₂×C₇×#67	1	2	yes	—
264	C₃	C₇×#537	1	1	yes	characteristic, complemented, Sylow
265	C₂	—	2	2	no	—
266	C₂	—	4	2	no	—
267	C₂	—	12	2	no	—
268	C₂	C₇×#1009	1	1	yes	characteristic
269	C₂	C₇×#1010	1	1	yes	characteristic
270	C₂	C₂²×C₇×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₇	C₂³×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×#537
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×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×#946	C₂	semidirect
4. C₇×#939	C₂	semidirect
5. C₇×#938	C₂	semidirect
6. C₇×#930	C₂	semidirect
7. C₂×C₇×C₄⋊Dic₃	C₂	semidirect
8. C₇×#924	C₂	semidirect
29. C₂×C₇×#41	D₃	semidirect
36. #5833	C₇	direct (N × H)
44. C₂×C₃×C₄×C₇	D₄	semidirect
74. C₂×C₃×#41	C₂×C₇	semidirect
75. C₂×#235	C₂×C₇	semidirect
76. #946	C₂×C₇	semidirect
77. #939	C₂×C₇	semidirect
78. #938	C₂×C₇	semidirect
79. #930	C₂×C₇	semidirect
80. C₂×C₄⋊Dic₃	C₂×C₇	semidirect
81. #924	C₂×C₇	semidirect
113. C₂×C₄×C₇	#67	semidirect
142. C₂×#41	D₃×C₇	semidirect
171. C₂×C₃×C₄	C₇×D₄	semidirect
183. C₃×C₇	#537	semidirect
238. C₂×C₄	C₇×#67	semidirect
242. C₇	#5833	direct (N × H)
264. C₃	C₇×#537	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×#946	2
C₇×#939	2
C₇×#938	2
C₇×#930	2
C₂×C₇×C₄⋊Dic₃	2
C₇×#924	2
C₇×#537	3
#5833	7

Subgroups — C7×#5833 #13061548

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#5833 #13061548