Subgroups — C₇×#12833 #13190049

Order 1792.

358 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3407	C₂	1	1	yes	maximal, characteristic
2	C₇×#2155	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×#365	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#1815	C₂	1	1	yes	maximal, characteristic
5	C₇×#1814	C₂	1	1	yes	maximal, characteristic
6	C₇×#1388	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1386	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₇×#131	C₂²	1	1	yes	characteristic
9	C₇×#504	—	2	2	no	—
10	C₇×#504	C₄	1	4	yes	complemented
11	C₂×C₇×M₃₂	—	2	2	no	—
12	C₂×C₇×M₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₇×M₃₂	C₂²	1	1	yes	characteristic
14	C₂×C₇×#108	C₂²	1	1	yes	characteristic
15	C₂×C₇×#108	C₂²	1	1	yes	characteristic
16	C₇×#427	C₂²	1	1	yes	characteristic
17	C₂×C₇×#106	C₂²	1	1	yes	characteristic
18	C₇×#421	—	2	1	no	—
19	C₇×#417	—	2	1	no	—
20	C₇×#365	—	2	2	no	—
21	C₇×#365	—	2	2	no	—
22	C₇×#344	—	2	1	no	—
23	C₇×#343	—	2	1	no	—
24	#12833	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂×C₇×#41	D₄	1	1	yes	characteristic
26	C₇×#131	—	2	1	no	—
27	C₇×#131	—	2	1	no	—
28	C₇×#131	C₂×C₄	1	2	yes	—
29	C₇×#131	Q₈	1	2	yes	—
30	C₂×C₇×M₁₆	—	2	1	no	—
31	C₂×C₇×M₁₆	—	2	1	no	—
32	C₂×C₇×M₁₆	—	2	1	no	—
33	C₂×C₇×M₁₆	D₄	1	1	yes	characteristic
34	C₂²×C₇×C₈	—	2	1	no	—
35	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
36	C₇×#117	—	2	1	no	—
37	C₂×C₇×C₄⋊C₄	—	2	1	no	—
38	C₇×M₃₂	—	2	1	no	—
39	C₇×M₃₂	—	2	4	no	—
40	C₇×M₃₂	C₂×C₄	1	2	yes	—
41	C₇×M₃₂	D₄	1	2	yes	—
42	C₇×M₃₂	D₄	1	2	yes	—
43	C₂×C₇×C₁₆	—	2	1	no	—
44	C₂×C₇×C₁₆	—	2	4	no	—
45	C₂×C₇×C₁₆	C₂×C₄	1	2	yes	—
46	C₇×#108	—	2	1	no	—
47	C₇×#108	—	2	1	no	—
48	C₇×#108	—	2	1	no	—
49	C₇×#108	—	2	1	no	—
50	C₇×#107	—	2	1	no	—
51	C₇×#106	—	2	1	no	—
52	C₇×#106	—	2	1	no	—
53	C₇×#106	—	2	1	no	—
54	C₇×#104	—	4	2	no	—
55	C₇×#103	—	4	1	no	—
56	C₇×#102	—	4	1	no	—
57	#3407	C₂×C₇	1	1	yes	characteristic
58	#2155	C₂×C₇	1	1	yes	characteristic, complemented
59	C₂×#365	C₂×C₇	1	1	yes	characteristic, complemented
60	#1815	C₂×C₇	1	1	yes	characteristic
61	#1814	C₂×C₇	1	1	yes	characteristic
62	#1388	C₂×C₇	1	1	yes	characteristic, complemented
63	#1386	C₂×C₇	1	1	yes	characteristic, complemented
64	C₇×#41	—	2	1	no	—
65	C₇×#41	—	2	1	no	—
66	C₇×#41	C₄⋊C₄	1	2	yes	—
67	C₇×#41	SD₁₆	1	2	yes	—
68	C₂×C₇×Q₈	SD₁₆	1	1	yes	characteristic
69	C₂×C₇×D₄	—	2	1	no	—
70	C₂×C₇×D₄	SD₁₆	1	1	yes	characteristic
71	C₂²×C₄×C₇	—	2	1	no	—
72	C₂²×C₄×C₇	—	4	1	no	—
73	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
74	C₇×M₁₆	—	2	1	no	—
75	C₇×M₁₆	—	2	2	no	—
76	C₇×M₁₆	—	2	2	no	—
77	C₇×M₁₆	—	4	1	no	—
78	C₇×M₁₆	—	4	1	no	—
79	C₇×M₁₆	—	4	1	no	—
80	C₇×M₁₆	—	4	1	no	—
81	C₇×M₁₆	C₄⋊C₄	1	2	yes	—
82	C₂×C₇×C₈	—	2	1	no	—
83	C₂×C₇×C₈	—	2	2	no	—
84	C₂×C₇×C₈	—	2	2	no	—
85	C₂×C₇×C₈	—	4	1	no	—
86	C₂×C₇×C₈	—	4	1	no	—
87	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
89	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
90	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
91	C₂×C₇×C₈	#41	1	1	yes	characteristic
92	C₂×C₇×C₈	#41	1	1	yes	characteristic
93	C₇×C₄⋊C₄	—	4	1	no	—
94	C₇×C₄⋊C₄	—	4	1	no	—
95	C₇×C₄⋊C₄	—	4	1	no	—
96	C₇×C₂²⋊C₄	—	4	1	no	—
97	C₄²×C₇	—	4	1	no	—
98	C₇×C₁₆	—	2	2	no	—
99	C₇×C₁₆	C₂²⋊C₄	1	4	yes	—
100	C₂×#131	C₂²×C₇	1	1	yes	characteristic
101	#504	—	2	2	no	—
102	#504	C₄×C₇	1	4	yes	complemented
103	C₂×M₃₂	—	2	2	no	—
104	C₂×M₃₂	C₂²×C₇	1	1	yes	characteristic
105	C₂×M₃₂	C₂²×C₇	1	1	yes	characteristic
106	C₂×#108	C₂²×C₇	1	1	yes	characteristic
107	C₂×#108	C₂²×C₇	1	1	yes	characteristic
108	#427	C₂²×C₇	1	1	yes	characteristic
109	C₂×#106	C₂²×C₇	1	1	yes	characteristic
110	#421	—	2	1	no	—
111	#417	—	2	1	no	—
112	#365	—	2	2	no	—
113	#365	—	2	2	no	—
114	#344	—	2	1	no	—
115	#343	—	2	1	no	—
116	C₂³×C₇	—	2	1	no	—
117	C₂³×C₇	#135	1	1	yes	characteristic
118	C₇×Q₈	—	2	1	no	—
119	C₇×Q₈	#106	1	2	yes	—
120	C₇×D₄	—	2	2	no	—
121	C₇×D₄	—	2	2	no	—
122	C₇×D₄	—	2	1	no	—
123	C₇×D₄	#106	1	2	yes	—
124	C₂×C₄×C₇	—	2	1	no	—
125	C₂×C₄×C₇	—	2	2	no	—
126	C₂×C₄×C₇	—	2	2	no	—
127	C₂×C₄×C₇	—	8	1	no	—
128	C₂×C₄×C₇	—	8	1	no	—
129	C₂×C₄×C₇	—	16	1	no	—
130	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₇	#121	1	1	yes	characteristic
132	C₂×C₄×C₇	#121	1	1	yes	characteristic
133	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
135	C₂×C₄×C₇	#135	1	1	yes	characteristic
136	C₇×C₈	—	2	2	no	—
137	C₇×C₈	—	8	1	no	—
138	C₇×C₈	—	8	1	no	—
139	C₇×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₇×C₈	#117	1	1	yes	characteristic
141	C₇×C₈	C₂²⋊Q₈	1	1	yes	characteristic
142	C₇×C₈	C₂²⋊Q₈	1	1	yes	characteristic
143	C₂×#41	C₇×D₄	1	1	yes	characteristic
144	#131	—	2	1	no	—
145	#131	—	2	1	no	—
146	#131	C₂×C₄×C₇	1	2	yes	—
147	#131	C₇×Q₈	1	2	yes	—
148	C₂×M₁₆	—	2	1	no	—
149	C₂×M₁₆	—	2	1	no	—
150	C₂×M₁₆	—	2	1	no	—
151	C₂×M₁₆	C₇×D₄	1	1	yes	characteristic
152	C₂²×C₈	—	2	1	no	—
153	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
154	#117	—	2	1	no	—
155	C₂×C₄⋊C₄	—	2	1	no	—
156	M₃₂	—	2	1	no	—
157	M₃₂	—	2	4	no	—
158	M₃₂	C₂×C₄×C₇	1	2	yes	—
159	M₃₂	C₇×D₄	1	2	yes	—
160	M₃₂	C₇×D₄	1	2	yes	—
161	C₂×C₁₆	—	2	1	no	—
162	C₂×C₁₆	—	2	4	no	—
163	C₂×C₁₆	C₂×C₄×C₇	1	2	yes	—
164	#108	—	2	1	no	—
165	#108	—	2	1	no	—
166	#108	—	2	1	no	—
167	#108	—	2	1	no	—
168	#107	—	2	1	no	—
169	#106	—	2	1	no	—
170	#106	—	2	1	no	—
171	#106	—	2	1	no	—
172	#104	—	4	2	no	—
173	#103	—	4	1	no	—
174	#102	—	4	1	no	—
175	C₂²×C₇	—	2	2	no	—
176	C₂²×C₇	—	4	1	no	—
177	C₂²×C₇	—	8	1	no	—
178	C₂²×C₇	#427	1	1	yes	characteristic
179	C₂²×C₇	#465	1	1	yes	characteristic
180	C₂²×C₇	#465	1	1	yes	characteristic
181	C₄×C₇	—	2	2	no	—
182	C₄×C₇	—	16	1	no	—
183	C₄×C₇	—	16	1	no	—
184	C₄×C₇	#380	1	1	yes	characteristic
185	C₄×C₇	C₂×#106	1	1	yes	characteristic
186	C₄×C₇	#465	1	1	yes	characteristic
187	C₄×C₇	#465	1	1	yes	characteristic
188	#41	—	2	1	no	—
189	#41	—	2	1	no	—
190	#41	C₇×C₄⋊C₄	1	2	yes	—
191	#41	C₇×SD₁₆	1	2	yes	—
192	C₂×Q₈	C₇×SD₁₆	1	1	yes	characteristic
193	C₂×D₄	—	2	1	no	—
194	C₂×D₄	C₇×SD₁₆	1	1	yes	characteristic
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	4	1	no	—
197	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
198	M₁₆	—	2	1	no	—
199	M₁₆	—	2	2	no	—
200	M₁₆	—	2	2	no	—
201	M₁₆	—	4	1	no	—
202	M₁₆	—	4	1	no	—
203	M₁₆	—	4	1	no	—
204	M₁₆	—	4	1	no	—
205	M₁₆	C₇×C₄⋊C₄	1	2	yes	—
206	C₂×C₈	—	2	1	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	4	1	no	—
210	C₂×C₈	—	4	1	no	—
211	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
212	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
213	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
214	C₂×C₈	C₂×C₇×Q₈	1	1	yes	characteristic
215	C₂×C₈	C₇×#41	1	1	yes	characteristic
216	C₂×C₈	C₇×#41	1	1	yes	characteristic
217	C₄⋊C₄	—	4	1	no	—
218	C₄⋊C₄	—	4	1	no	—
219	C₄⋊C₄	—	4	1	no	—
220	C₂²⋊C₄	—	4	1	no	—
221	C₄²	—	4	1	no	—
222	C₁₆	—	2	2	no	—
223	C₁₆	C₇×C₂²⋊C₄	1	4	yes	—
224	C₂×C₇	—	2	1	no	—
225	C₂×C₇	—	2	1	no	—
226	C₂×C₇	—	2	1	no	—
227	C₂×C₇	—	4	2	no	—
228	C₂×C₇	#1807	1	1	yes	characteristic
229	C₂³	—	2	1	no	—
230	C₂³	C₇×#135	1	1	yes	characteristic
231	Q₈	—	2	1	no	—
232	Q₈	C₇×#106	1	2	yes	—
233	D₄	—	2	2	no	—
234	D₄	—	2	2	no	—
235	D₄	—	2	1	no	—
236	D₄	C₇×#106	1	2	yes	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	8	1	no	—
241	C₂×C₄	—	8	1	no	—
242	C₂×C₄	—	16	1	no	—
243	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
244	C₂×C₄	C₇×#121	1	1	yes	characteristic
245	C₂×C₄	C₇×#121	1	1	yes	characteristic
246	C₂×C₄	C₂×C₇×SD₁₆	1	1	yes	characteristic
247	C₂×C₄	C₂×C₇×SD₁₆	1	1	yes	characteristic
248	C₂×C₄	C₇×#135	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	—	8	1	no	—
251	C₈	—	8	1	no	—
252	C₈	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
253	C₈	C₇×#117	1	1	yes	characteristic
254	C₈	C₇×C₂²⋊Q₈	1	1	yes	characteristic
255	C₈	C₇×C₂²⋊Q₈	1	1	yes	characteristic
256	C₇	#12833	1	1	yes	characteristic, direct product factor, Sylow
257	C₂²	—	2	2	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	8	1	no	—
260	C₂²	C₇×#427	1	1	yes	characteristic
261	C₂²	C₇×#465	1	1	yes	characteristic
262	C₂²	C₇×#465	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	16	1	no	—
265	C₄	—	16	1	no	—
266	C₄	C₇×#380	1	1	yes	characteristic
267	C₄	C₂×C₇×#106	1	1	yes	characteristic
268	C₄	C₇×#465	1	1	yes	characteristic
269	C₄	C₇×#465	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	2	1	no	—
272	C₂	—	2	1	no	—
273	C₂	—	4	2	no	—
274	C₂	C₇×#1807	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₇	#1807
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₂²×C₈	C₂³×C₇
prime radical	C₂×C₇×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₇×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#2155	C₂	semidirect
3. C₂×C₇×#365	C₂	semidirect
6. C₇×#1388	C₂	semidirect
7. C₇×#1386	C₂	semidirect
10. C₇×#504	C₄	semidirect
24. #12833	C₇	direct (N × H)
58. #2155	C₂×C₇	semidirect
59. C₂×#365	C₂×C₇	semidirect
62. #1388	C₂×C₇	semidirect
63. #1386	C₂×C₇	semidirect
102. #504	C₄×C₇	semidirect
256. C₇	#12833	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#3407	2
C₇×#2155	2
C₂×C₇×#365	2
C₇×#1815	2
C₇×#1814	2
C₇×#1388	2
C₇×#1386	2
#12833	7

Subgroups — C7×#12833 #13190049

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#12833 #13190049