Subgroups — C₇×#12818 #13190034

Order 1792.

474 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×D₈	C₂	1	1	yes	maximal, characteristic
2	C₇×#2150	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×#361	C₂	1	2	yes	maximal
4	C₇×#1848	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1387	C₂	1	2	yes	maximal, complemented
6	C₂²×C₇×D₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₇×D₄	—	2	1	no	—
8	C₂×C₇×M₃₂	C₂²	1	2	yes	—
9	C₄×C₇×D₈	—	2	1	no	—
10	C₄×C₇×D₈	—	2	1	no	—
11	C₄×C₇×D₈	C₄	1	2	yes	—
12	C₄×C₇×D₈	C₄	1	2	yes	—
13	C₂×C₇×#108	—	2	1	no	—
14	C₂×C₇×#108	C₂²	1	2	yes	—
15	C₂×C₇×#107	C₂²	1	1	yes	characteristic
16	C₇×#423	—	2	1	no	—
17	C₂×C₇×#102	—	2	1	no	—
18	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
19	C₇×C₄⋊C₁₆	—	2	1	no	—
20	C₇×C₄⋊C₁₆	—	2	1	no	—
21	C₇×#361	—	2	4	no	—
22	#12818	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂²×C₇×D₄	—	2	1	no	—
24	C₂³×C₄×C₇	—	4	1	no	—
25	C₂×C₇×D₈	—	2	1	no	—
26	C₂×C₇×D₈	—	2	1	no	—
27	C₂×C₇×D₈	—	2	1	no	—
28	C₂×C₇×D₈	C₂×C₄	1	2	yes	—
29	C₂×C₇×D₈	D₄	1	2	yes	—
30	C₂×C₇×D₈	Q₈	1	2	yes	—
31	C₂×C₇×M₁₆	—	2	2	no	—
32	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
33	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
34	C₄×C₇×D₄	—	2	2	no	—
35	C₄×C₇×D₄	—	2	2	no	—
36	C₄×C₇×D₄	—	4	1	no	—
37	C₄×C₇×D₄	—	4	1	no	—
38	C₂×C₇×C₄⋊C₄	—	2	1	no	—
39	C₂×C₇×C₂²⋊C₄	—	4	1	no	—
40	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
41	C₇×M₃₂	—	2	2	no	—
42	C₇×M₃₂	—	2	2	no	—
43	C₂×C₇×C₁₆	—	2	2	no	—
44	C₇×#108	—	2	2	no	—
45	C₇×#108	—	2	2	no	—
46	C₇×#108	—	4	2	no	—
47	C₇×#107	—	2	1	no	—
48	C₇×#107	C₂×C₄	1	2	yes	—
49	C₇×C₄⋊C₈	—	4	1	no	—
50	C₇×C₄⋊C₈	—	4	1	no	—
51	C₇×#102	—	2	2	no	—
52	C₇×#102	—	2	2	no	—
53	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
54	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
55	C₄×C₇×C₈	D₄	1	1	yes	characteristic
56	C₄×C₇×C₈	D₄	1	1	yes	characteristic
57	C₂×C₄×D₈	C₂×C₇	1	1	yes	characteristic
58	#2150	C₂×C₇	1	1	yes	characteristic, complemented
59	C₂×#361	C₂×C₇	1	2	yes	—
60	#1848	C₂×C₇	1	1	yes	characteristic, complemented
61	#1387	C₂×C₇	1	2	yes	complemented
62	C₂⁴×C₇	—	4	1	no	—
63	C₂×C₇×D₄	—	2	2	no	—
64	C₂×C₇×D₄	—	2	2	no	—
65	C₂×C₇×D₄	—	2	2	no	—
66	C₂×C₇×D₄	—	4	1	no	—
67	C₂×C₇×D₄	—	4	1	no	—
68	C₂×C₇×D₄	—	4	1	no	—
69	C₂²×C₄×C₇	—	4	1	no	—
70	C₂²×C₄×C₇	—	4	2	no	—
71	C₂²×C₄×C₇	—	4	2	no	—
72	C₂²×C₄×C₇	—	8	4	no	—
73	C₂²×C₄×C₇	SD₁₆	1	1	yes	characteristic
74	C₂²×C₄×C₇	SD₁₆	1	1	yes	characteristic
75	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
76	C₇×D₈	—	2	2	no	—
77	C₇×D₈	—	2	2	no	—
78	C₇×D₈	—	2	2	no	—
79	C₇×D₈	C₄⋊C₄	1	4	yes	—
80	C₇×M₁₆	—	4	2	no	—
81	C₇×M₁₆	—	4	2	no	—
82	C₂×C₇×C₈	—	2	1	no	—
83	C₂×C₇×C₈	—	2	1	no	—
84	C₂×C₇×C₈	—	4	2	no	—
85	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
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	—	2	2	no	—
94	C₇×C₄⋊C₄	—	4	1	no	—
95	C₇×C₂²⋊C₄	—	4	2	no	—
96	C₇×C₂²⋊C₄	—	4	2	no	—
97	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
98	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
99	C₄²×C₇	D₈	1	1	yes	characteristic
100	C₄²×C₇	D₈	1	1	yes	characteristic
101	C₇×C₁₆	—	4	2	no	—
102	C₂²×D₈	C₂²×C₇	1	1	yes	characteristic
103	C₂×C₄×D₄	—	2	1	no	—
104	C₂×M₃₂	C₂²×C₇	1	2	yes	—
105	C₄×D₈	—	2	1	no	—
106	C₄×D₈	—	2	1	no	—
107	C₄×D₈	C₄×C₇	1	2	yes	—
108	C₄×D₈	C₄×C₇	1	2	yes	—
109	C₂×#108	—	2	1	no	—
110	C₂×#108	C₂²×C₇	1	2	yes	—
111	C₂×#107	C₂²×C₇	1	1	yes	characteristic
112	#423	—	2	1	no	—
113	C₂×#102	—	2	1	no	—
114	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
115	C₄⋊C₁₆	—	2	1	no	—
116	C₄⋊C₁₆	—	2	1	no	—
117	#361	—	2	4	no	—
118	C₂³×C₇	—	4	2	no	—
119	C₂³×C₇	—	4	2	no	—
120	C₂³×C₇	—	4	2	no	—
121	C₂³×C₇	—	8	2	no	—
122	C₂³×C₇	—	8	2	no	—
123	C₂³×C₇	C₂×SD₁₆	1	1	yes	characteristic
124	C₇×D₄	—	2	4	no	—
125	C₇×D₄	—	4	2	no	—
126	C₇×D₄	—	4	2	no	—
127	C₇×D₄	—	4	2	no	—
128	C₂×C₄×C₇	—	2	2	no	—
129	C₂×C₄×C₇	—	4	1	no	—
130	C₂×C₄×C₇	—	4	2	no	—
131	C₂×C₄×C₇	—	8	4	no	—
132	C₂×C₄×C₇	—	8	2	no	—
133	C₂×C₄×C₇	—	8	4	no	—
134	C₂×C₄×C₇	#102	1	1	yes	characteristic
135	C₂×C₄×C₇	#102	1	1	yes	characteristic
136	C₂×C₄×C₇	#102	1	1	yes	characteristic
137	C₂×C₄×C₇	#102	1	1	yes	characteristic
138	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₇	#121	1	1	yes	characteristic
140	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
141	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂×C₄×C₇	#137	1	1	yes	characteristic
144	C₇×C₈	—	2	1	no	—
145	C₇×C₈	—	2	1	no	—
146	C₇×C₈	—	8	2	no	—
147	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₇×C₈	#117	1	1	yes	characteristic
149	C₇×C₈	#121	1	1	yes	characteristic
150	C₇×C₈	C₂²⋊Q₈	1	1	yes	characteristic
151	C₂²×D₄	—	2	1	no	—
152	C₂³×C₄	—	4	1	no	—
153	C₂×D₈	—	2	1	no	—
154	C₂×D₈	—	2	1	no	—
155	C₂×D₈	—	2	1	no	—
156	C₂×D₈	C₂×C₄×C₇	1	2	yes	—
157	C₂×D₈	C₇×D₄	1	2	yes	—
158	C₂×D₈	C₇×Q₈	1	2	yes	—
159	C₂×M₁₆	—	2	2	no	—
160	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
161	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
162	C₄×D₄	—	2	2	no	—
163	C₄×D₄	—	2	2	no	—
164	C₄×D₄	—	4	1	no	—
165	C₄×D₄	—	4	1	no	—
166	C₂×C₄⋊C₄	—	2	1	no	—
167	C₂×C₂²⋊C₄	—	4	1	no	—
168	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
169	M₃₂	—	2	2	no	—
170	M₃₂	—	2	2	no	—
171	C₂×C₁₆	—	2	2	no	—
172	#108	—	2	2	no	—
173	#108	—	2	2	no	—
174	#108	—	4	2	no	—
175	#107	—	2	1	no	—
176	#107	C₂×C₄×C₇	1	2	yes	—
177	C₄⋊C₈	—	4	1	no	—
178	C₄⋊C₈	—	4	1	no	—
179	#102	—	2	2	no	—
180	#102	—	2	2	no	—
181	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
182	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
183	C₄×C₈	C₇×D₄	1	1	yes	characteristic
184	C₄×C₈	C₇×D₄	1	1	yes	characteristic
185	C₂²×C₇	—	2	1	no	—
186	C₂²×C₇	—	2	1	no	—
187	C₂²×C₇	—	4	4	no	—
188	C₂²×C₇	—	8	4	no	—
189	C₂²×C₇	—	8	4	no	—
190	C₂²×C₇	—	8	2	no	—
191	C₂²×C₇	—	8	2	no	—
192	C₂²×C₇	C₂×#102	1	1	yes	characteristic
193	C₂²×C₇	#461	1	1	yes	characteristic
194	C₂²×C₇	#476	1	1	yes	characteristic
195	C₄×C₇	—	2	1	no	—
196	C₄×C₇	—	2	1	no	—
197	C₄×C₇	—	2	1	no	—
198	C₄×C₇	—	8	2	no	—
199	C₄×C₇	#361	1	2	yes	complemented
200	C₄×C₇	#380	1	1	yes	characteristic
201	C₄×C₇	#417	1	1	yes	characteristic
202	C₄×C₇	#459	1	1	yes	characteristic
203	C₄×C₇	#474	1	1	yes	characteristic
204	C₂⁴	—	4	1	no	—
205	C₂×D₄	—	2	2	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	—	4	1	no	—
209	C₂×D₄	—	4	1	no	—
210	C₂×D₄	—	4	1	no	—
211	C₂²×C₄	—	4	1	no	—
212	C₂²×C₄	—	4	2	no	—
213	C₂²×C₄	—	4	2	no	—
214	C₂²×C₄	—	8	4	no	—
215	C₂²×C₄	C₇×SD₁₆	1	1	yes	characteristic
216	C₂²×C₄	C₇×SD₁₆	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
218	D₈	—	2	2	no	—
219	D₈	—	2	2	no	—
220	D₈	—	2	2	no	—
221	D₈	C₇×C₄⋊C₄	1	4	yes	—
222	M₁₆	—	4	2	no	—
223	M₁₆	—	4	2	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
228	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
229	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
230	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
231	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
232	C₂×C₈	C₂×C₇×Q₈	1	1	yes	characteristic
233	C₂×C₈	C₇×#41	1	1	yes	characteristic
234	C₂×C₈	C₇×#41	1	1	yes	characteristic
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	4	1	no	—
237	C₂²⋊C₄	—	4	2	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
240	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
241	C₄²	C₇×D₈	1	1	yes	characteristic
242	C₄²	C₇×D₈	1	1	yes	characteristic
243	C₁₆	—	4	2	no	—
244	C₂×C₇	—	2	1	no	—
245	C₂×C₇	—	2	1	no	—
246	C₂×C₇	—	8	4	no	—
247	C₂×C₇	#1802	1	1	yes	characteristic
248	C₂×C₇	C₂×#361	1	1	yes	characteristic
249	C₂×C₇	#2148	1	1	yes	characteristic
250	C₂³	—	4	2	no	—
251	C₂³	—	4	2	no	—
252	C₂³	—	4	2	no	—
253	C₂³	—	8	2	no	—
254	C₂³	—	8	2	no	—
255	C₂³	C₂×C₇×SD₁₆	1	1	yes	characteristic
256	D₄	—	2	4	no	—
257	D₄	—	4	2	no	—
258	D₄	—	4	2	no	—
259	D₄	—	4	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	8	4	no	—
264	C₂×C₄	—	8	2	no	—
265	C₂×C₄	—	8	4	no	—
266	C₂×C₄	C₇×#102	1	1	yes	characteristic
267	C₂×C₄	C₇×#102	1	1	yes	characteristic
268	C₂×C₄	C₇×#102	1	1	yes	characteristic
269	C₂×C₄	C₇×#102	1	1	yes	characteristic
270	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
271	C₂×C₄	C₇×#121	1	1	yes	characteristic
272	C₂×C₄	C₇×C₂²⋊Q₈	1	1	yes	characteristic
273	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
274	C₂×C₄	C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
275	C₂×C₄	C₇×#137	1	1	yes	characteristic
276	C₈	—	2	1	no	—
277	C₈	—	2	1	no	—
278	C₈	—	8	2	no	—
279	C₈	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
280	C₈	C₇×#117	1	1	yes	characteristic
281	C₈	C₇×#121	1	1	yes	characteristic
282	C₈	C₇×C₂²⋊Q₈	1	1	yes	characteristic
283	C₇	#12818	1	1	yes	characteristic, direct product factor, Sylow
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	4	4	no	—
287	C₂²	—	8	4	no	—
288	C₂²	—	8	4	no	—
289	C₂²	—	8	2	no	—
290	C₂²	—	8	2	no	—
291	C₂²	C₂×C₇×#102	1	1	yes	characteristic
292	C₂²	C₇×#461	1	1	yes	characteristic
293	C₂²	C₇×#476	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	8	2	no	—
298	C₄	C₇×#361	1	2	yes	complemented
299	C₄	C₇×#380	1	1	yes	characteristic
300	C₄	C₇×#417	1	1	yes	characteristic
301	C₄	C₇×#459	1	1	yes	characteristic
302	C₄	C₇×#474	1	1	yes	characteristic
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	8	4	no	—
306	C₂	C₇×#1802	1	1	yes	characteristic
307	C₂	C₂×C₇×#361	1	1	yes	characteristic
308	C₂	C₇×#2148	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₂²×C₈	C₂³×C₇
prime radical	C₂²×C₇×D₈	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×#1848	C₂	semidirect
5. C₇×#1387	C₂	semidirect
22. #12818	C₇	direct (N × H)
58. #2150	C₂×C₇	semidirect
60. #1848	C₂×C₇	semidirect
61. #1387	C₂×C₇	semidirect
199. C₄×C₇	#361	semidirect
283. C₇	#12818	direct (N × H)
298. C₄	C₇×#361	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₇×D₈	2
C₇×#2150	2
C₂×C₇×#361	2
C₇×#1848	2
C₇×#1387	2
#12818	7

Subgroups — C7×#12818 #13190034

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#12818 #13190034