Subgroups — C₇×#12817 #13190033

Order 1792.

430 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×#135	C₂	1	1	yes	maximal, characteristic
2	C₇×#2150	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×#360	C₂	1	2	yes	maximal
4	C₇×#1846	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1386	C₂	1	2	yes	maximal, complemented
6	C₂×C₇×#135	C₂²	1	1	yes	characteristic
7	C₄×C₇×#41	—	2	1	no	—
8	C₂×C₇×M₃₂	C₂²	1	2	yes	—
9	C₄×C₇×Q₁₆	C₄	1	2	yes	complemented
10	C₄×C₇×SD₁₆	—	2	2	no	—
11	C₄×C₇×D₈	C₄	1	2	yes	complemented
12	C₇×#427	C₂²	1	1	yes	characteristic
13	C₂×C₇×#107	—	2	1	no	—
14	C₂×C₇×#106	C₂²	1	2	yes	—
15	C₇×#416	—	2	1	no	—
16	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
17	C₇×#382	—	2	1	no	—
18	C₇×C₄⋊C₁₆	—	2	2	no	—
19	C₇×#360	—	2	4	no	—
20	#12817	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂×C₇×#41	—	2	1	no	—
22	C₇×#135	—	2	1	no	—
23	C₇×#135	—	2	1	no	—
24	C₇×#135	D₄	1	2	yes	—
25	C₇×#135	Q₈	1	2	yes	—
26	C₂×C₇×Q₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂×C₇×SD₁₆	—	2	1	no	—
28	C₂×C₇×D₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
30	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
31	C₄×C₇×Q₈	—	2	2	no	—
32	C₄×C₇×D₄	—	2	2	no	—
33	C₄×C₇×D₄	—	4	2	no	—
34	C₇×#117	—	2	1	no	—
35	C₇×#117	—	4	1	no	—
36	C₂×C₇×C₄⋊C₄	—	2	2	no	—
37	C₂×C₄²×C₇	—	4	1	no	—
38	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
39	C₇×M₃₂	—	2	2	no	—
40	C₇×M₃₂	—	2	2	no	—
41	C₂×C₇×C₁₆	—	2	2	no	—
42	C₇×#107	—	4	2	no	—
43	C₇×#107	C₂×C₄	1	2	yes	—
44	C₇×#106	—	2	1	no	—
45	C₇×#106	—	2	2	no	—
46	C₇×#106	—	2	2	no	—
47	C₇×#103	—	2	2	no	—
48	C₇×#102	—	2	2	no	—
49	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
50	C₄×C₇×C₈	D₄	1	2	yes	—
51	C₇×#95	—	4	2	no	—
52	C₄×#135	C₂×C₇	1	1	yes	characteristic
53	#2150	C₂×C₇	1	1	yes	characteristic, complemented
54	C₂×#360	C₂×C₇	1	2	yes	—
55	#1846	C₂×C₇	1	1	yes	characteristic, complemented
56	#1386	C₂×C₇	1	2	yes	complemented
57	C₇×#41	—	2	2	no	—
58	C₇×#41	—	2	2	no	—
59	C₇×#41	—	4	1	no	—
60	C₇×#41	—	4	1	no	—
61	C₂×C₇×Q₈	—	2	1	no	—
62	C₂×C₇×D₄	—	2	1	no	—
63	C₂×C₇×D₄	—	4	1	no	—
64	C₂²×C₄×C₇	—	4	1	no	—
65	C₂²×C₄×C₇	—	4	2	no	—
66	C₂²×C₄×C₇	—	4	2	no	—
67	C₂²×C₄×C₇	D₈	1	2	yes	—
68	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
69	C₇×Q₁₆	—	2	1	no	—
70	C₇×Q₁₆	C₄⋊C₄	1	2	yes	—
71	C₇×SD₁₆	—	2	2	no	—
72	C₇×SD₁₆	—	2	2	no	—
73	C₇×D₈	—	2	1	no	—
74	C₇×D₈	C₄⋊C₄	1	2	yes	—
75	C₂×C₇×C₈	—	2	1	no	—
76	C₂×C₇×C₈	—	2	1	no	—
77	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
78	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
80	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
81	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
82	C₂×C₇×C₈	#41	1	1	yes	characteristic
83	C₂×C₇×C₈	#41	1	1	yes	characteristic
84	C₇×C₄⋊C₄	—	2	2	no	—
85	C₇×C₄⋊C₄	—	4	2	no	—
86	C₇×C₄⋊C₄	—	4	2	no	—
87	C₇×C₄⋊C₄	—	4	2	no	—
88	C₇×C₂²⋊C₄	—	4	1	no	—
89	C₇×C₂²⋊C₄	—	4	2	no	—
90	C₄²×C₇	—	4	2	no	—
91	C₄²×C₇	—	4	1	no	—
92	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
93	C₄²×C₇	SD₁₆	1	1	yes	characteristic
94	C₄²×C₇	SD₁₆	1	1	yes	characteristic
95	C₇×C₁₆	—	4	2	no	—
96	C₂×#135	C₂²×C₇	1	1	yes	characteristic
97	C₄×#41	—	2	1	no	—
98	C₂×M₃₂	C₂²×C₇	1	2	yes	—
99	C₄×Q₁₆	C₄×C₇	1	2	yes	complemented
100	C₄×SD₁₆	—	2	2	no	—
101	C₄×D₈	C₄×C₇	1	2	yes	complemented
102	#427	C₂²×C₇	1	1	yes	characteristic
103	C₂×#107	—	2	1	no	—
104	C₂×#106	C₂²×C₇	1	2	yes	—
105	#416	—	2	1	no	—
106	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
107	#382	—	2	1	no	—
108	C₄⋊C₁₆	—	2	2	no	—
109	#360	—	2	4	no	—
110	C₂³×C₇	—	4	1	no	—
111	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
112	C₇×Q₈	—	2	2	no	—
113	C₇×Q₈	—	4	1	no	—
114	C₇×D₄	—	2	2	no	—
115	C₇×D₄	—	4	2	no	—
116	C₇×D₄	—	4	2	no	—
117	C₇×D₄	—	4	1	no	—
118	C₂×C₄×C₇	—	4	2	no	—
119	C₂×C₄×C₇	—	4	1	no	—
120	C₂×C₄×C₇	—	4	2	no	—
121	C₂×C₄×C₇	—	4	2	no	—
122	C₂×C₄×C₇	—	4	2	no	—
123	C₂×C₄×C₇	—	8	4	no	—
124	C₂×C₄×C₇	—	8	4	no	—
125	C₂×C₄×C₇	—	8	2	no	—
126	C₂×C₄×C₇	#102	1	2	yes	—
127	C₂×C₄×C₇	#102	1	2	yes	—
128	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₇	#121	1	1	yes	characteristic
130	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
131	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
132	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂×C₄×C₇	#137	1	1	yes	characteristic
134	C₇×C₈	—	2	2	no	—
135	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₇×C₈	#117	1	1	yes	characteristic
137	C₇×C₈	#121	1	1	yes	characteristic
138	C₇×C₈	C₂²⋊Q₈	1	1	yes	characteristic
139	C₂×#41	—	2	1	no	—
140	#135	—	2	1	no	—
141	#135	—	2	1	no	—
142	#135	C₇×D₄	1	2	yes	—
143	#135	C₇×Q₈	1	2	yes	—
144	C₂×Q₁₆	C₂×C₄×C₇	1	1	yes	characteristic
145	C₂×SD₁₆	—	2	1	no	—
146	C₂×D₈	C₂×C₄×C₇	1	1	yes	characteristic
147	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
148	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
149	C₄×Q₈	—	2	2	no	—
150	C₄×D₄	—	2	2	no	—
151	C₄×D₄	—	4	2	no	—
152	#117	—	2	1	no	—
153	#117	—	4	1	no	—
154	C₂×C₄⋊C₄	—	2	2	no	—
155	C₂×C₄²	—	4	1	no	—
156	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
157	M₃₂	—	2	2	no	—
158	M₃₂	—	2	2	no	—
159	C₂×C₁₆	—	2	2	no	—
160	#107	—	4	2	no	—
161	#107	C₂×C₄×C₇	1	2	yes	—
162	#106	—	2	1	no	—
163	#106	—	2	2	no	—
164	#106	—	2	2	no	—
165	#103	—	2	2	no	—
166	#102	—	2	2	no	—
167	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
168	C₄×C₈	C₇×D₄	1	2	yes	—
169	#95	—	4	2	no	—
170	C₂²×C₇	—	2	2	no	—
171	C₂²×C₇	—	4	2	no	—
172	C₂²×C₇	—	8	2	no	—
173	C₂²×C₇	C₂×#102	1	1	yes	characteristic
174	C₂²×C₇	#459	1	1	yes	characteristic
175	C₂²×C₇	#474	1	1	yes	characteristic
176	C₄×C₇	—	2	2	no	—
177	C₄×C₇	—	2	2	no	—
178	C₄×C₇	—	4	2	no	—
179	C₄×C₇	—	8	2	no	—
180	C₄×C₇	—	16	2	no	—
181	C₄×C₇	#380	1	1	yes	characteristic
182	C₄×C₇	#417	1	1	yes	characteristic
183	C₄×C₇	#461	1	1	yes	characteristic
184	C₄×C₇	#476	1	1	yes	characteristic
185	#41	—	2	2	no	—
186	#41	—	2	2	no	—
187	#41	—	4	1	no	—
188	#41	—	4	1	no	—
189	C₂×Q₈	—	2	1	no	—
190	C₂×D₄	—	2	1	no	—
191	C₂×D₄	—	4	1	no	—
192	C₂²×C₄	—	4	1	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	C₇×D₈	1	2	yes	—
196	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
197	Q₁₆	—	2	1	no	—
198	Q₁₆	C₇×C₄⋊C₄	1	2	yes	—
199	SD₁₆	—	2	2	no	—
200	SD₁₆	—	2	2	no	—
201	D₈	—	2	1	no	—
202	D₈	C₇×C₄⋊C₄	1	2	yes	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	C₇×C₂²⋊C₄	1	2	yes	—
206	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
207	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
208	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
209	C₂×C₈	C₂×C₇×Q₈	1	1	yes	characteristic
210	C₂×C₈	C₇×#41	1	1	yes	characteristic
211	C₂×C₈	C₇×#41	1	1	yes	characteristic
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	4	2	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₂²⋊C₄	—	4	1	no	—
217	C₂²⋊C₄	—	4	2	no	—
218	C₄²	—	4	2	no	—
219	C₄²	—	4	1	no	—
220	C₄²	C₇×C₂²⋊C₄	1	2	yes	—
221	C₄²	C₇×SD₁₆	1	1	yes	characteristic
222	C₄²	C₇×SD₁₆	1	1	yes	characteristic
223	C₁₆	—	4	2	no	—
224	C₂×C₇	—	2	1	no	—
225	C₂×C₇	—	2	1	no	—
226	C₂×C₇	—	8	2	no	—
227	C₂×C₇	#1802	1	1	yes	characteristic
228	C₂×C₇	#2145	1	2	yes	—
229	C₂³	—	4	1	no	—
230	C₂³	C₂×C₇×D₈	1	1	yes	characteristic
231	Q₈	—	2	2	no	—
232	Q₈	—	4	1	no	—
233	D₄	—	2	2	no	—
234	D₄	—	4	2	no	—
235	D₄	—	4	2	no	—
236	D₄	—	4	1	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	1	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	8	4	no	—
243	C₂×C₄	—	8	4	no	—
244	C₂×C₄	—	8	2	no	—
245	C₂×C₄	C₇×#102	1	2	yes	—
246	C₂×C₄	C₇×#102	1	2	yes	—
247	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
248	C₂×C₄	C₇×#121	1	1	yes	characteristic
249	C₂×C₄	C₇×C₂²⋊Q₈	1	1	yes	characteristic
250	C₂×C₄	C₂×C₇×SD₁₆	1	1	yes	characteristic
251	C₂×C₄	C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
252	C₂×C₄	C₇×#137	1	1	yes	characteristic
253	C₈	—	2	2	no	—
254	C₈	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
255	C₈	C₇×#117	1	1	yes	characteristic
256	C₈	C₇×#121	1	1	yes	characteristic
257	C₈	C₇×C₂²⋊Q₈	1	1	yes	characteristic
258	C₇	#12817	1	1	yes	characteristic, direct product factor, Sylow
259	C₂²	—	2	2	no	—
260	C₂²	—	4	2	no	—
261	C₂²	—	8	2	no	—
262	C₂²	C₂×C₇×#102	1	1	yes	characteristic
263	C₂²	C₇×#459	1	1	yes	characteristic
264	C₂²	C₇×#474	1	1	yes	characteristic
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	4	2	no	—
268	C₄	—	8	2	no	—
269	C₄	—	16	2	no	—
270	C₄	C₇×#380	1	1	yes	characteristic
271	C₄	C₇×#417	1	1	yes	characteristic
272	C₄	C₇×#461	1	1	yes	characteristic
273	C₄	C₇×#476	1	1	yes	characteristic
274	C₂	—	2	1	no	—
275	C₂	—	2	1	no	—
276	C₂	—	8	2	no	—
277	C₂	C₇×#1802	1	1	yes	characteristic
278	C₂	C₇×#2145	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₂²×C₈	C₂³×C₇
prime radical	C₂×C₇×#135	C₂²
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
2. C₇×#2150	C₂	semidirect
4. C₇×#1846	C₂	semidirect
5. C₇×#1386	C₂	semidirect
9. C₄×C₇×Q₁₆	C₄	semidirect
11. C₄×C₇×D₈	C₄	semidirect
20. #12817	C₇	direct (N × H)
53. #2150	C₂×C₇	semidirect
55. #1846	C₂×C₇	semidirect
56. #1386	C₂×C₇	semidirect
99. C₄×Q₁₆	C₄×C₇	semidirect
101. C₄×D₈	C₄×C₇	semidirect
258. C₇	#12817	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×#135	2
C₇×#2150	2
C₂×C₇×#360	2
C₇×#1846	2
C₇×#1386	2
#12817	7

Subgroups — C7×#12817 #13190033

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#12817 #13190033