Subgroups — C₇×Q₈×#103 #13199736

Order 1792.

1086 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×C₇×Q₈²	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
3	C₇×Q₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₇×#1990	C₂	1	3	yes	maximal
5	C₇×#1987	C₂	1	3	yes	maximal
6	C₇×#1803	C₂	1	3	yes	maximal
7	C₄×C₇×#103	C₂	1	3	yes	maximal
8	C₇×Q₈²	—	2	4	no	—
9	C₇×Q₈²	C₄	1	8	yes	complemented
10	C₂×C₇×C₄⋊Q₈	—	2	3	no	—
11	C₂×C₇×C₄⋊Q₈	C₂²	1	3	yes	—
12	C₂×C₄×C₇×Q₈	—	2	1	no	—
13	C₂×C₄×C₇×Q₈	—	2	1	no	—
14	C₂×C₄×C₇×Q₈	C₂²	1	3	yes	—
15	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
16	C₇×C₈×Q₈	—	2	4	no	—
17	C₂×C₇×C₄⋊C₈	C₂²	1	3	yes	—
18	C₂×C₇×#103	C₂²	1	3	yes	—
19	C₂×C₇×#103	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₇×C₈	C₂²	1	3	yes	—
21	C₇×#389	—	2	3	no	—
22	C₇×#384	C₂²	1	3	yes	—
23	C₄×C₇×C₄⋊C₄	C₂²	1	3	yes	—
24	C₇×#340	C₂²	1	3	yes	—
25	C₇×#340	C₂²	1	6	yes	—
26	C₇×#340	C₂²	1	3	yes	—
27	Q₈×#103	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂²×C₇×Q₈	—	2	3	no	—
29	C₂²×C₇×Q₈	D₄	1	1	yes	characteristic
30	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
31	C₂²×C₇×C₈	C₂³	1	3	yes	—
32	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
33	C₇×C₄⋊Q₈	—	2	24	no	—
34	C₇×C₄⋊Q₈	—	2	6	no	—
35	C₇×C₄⋊Q₈	C₂×C₄	1	12	yes	—
36	C₄×C₇×Q₈	—	2	8	no	—
37	C₄×C₇×Q₈	—	2	6	no	—
38	C₄×C₇×Q₈	—	4	4	no	—
39	C₄×C₇×Q₈	C₂×C₄	1	12	yes	—
40	C₄×C₇×Q₈	C₂×C₄	1	4	yes	—
41	C₄×C₇×Q₈	D₄	1	4	yes	—
42	C₂×C₇×C₄⋊C₄	—	2	3	no	—
43	C₂×C₇×C₄⋊C₄	—	2	3	no	—
44	C₂×C₇×C₄⋊C₄	—	2	3	no	—
45	C₂×C₇×C₄⋊C₄	—	2	3	no	—
46	C₂×C₇×C₄⋊C₄	D₄	1	3	yes	—
47	C₂×C₇×C₄⋊C₄	C₂³	1	3	yes	—
48	C₂×C₇×C₄⋊C₄	C₂³	1	3	yes	—
49	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₇	—	2	3	no	—
51	C₂×C₄²×C₇	—	2	3	no	—
52	C₂×C₄²×C₇	C₂³	1	3	yes	—
53	C₇×C₄⋊C₈	—	2	6	no	—
54	C₇×#103	—	2	6	no	—
55	C₇×#103	Q₈	1	4	yes	direct product factor
56	C₄×C₇×C₈	—	2	6	no	—
57	C₇×#95	—	2	3	no	—
58	C₂×Q₈²	C₂×C₇	1	1	yes	characteristic
59	C₂×C₈×Q₈	C₂×C₇	1	1	yes	characteristic
60	Q₈×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic
61	#1990	C₂×C₇	1	3	yes	—
62	#1987	C₂×C₇	1	3	yes	—
63	#1803	C₂×C₇	1	3	yes	—
64	C₄×#103	C₂×C₇	1	3	yes	—
65	C₂×C₇×Q₈	—	2	12	no	—
66	C₂×C₇×Q₈	—	2	1	no	—
67	C₂×C₇×Q₈	—	2	1	no	—
68	C₂×C₇×Q₈	—	2	1	no	—
69	C₂×C₇×Q₈	—	8	3	no	—
70	C₂×C₇×Q₈	—	8	3	no	—
71	C₂×C₇×Q₈	C₂²⋊C₄	1	4	yes	—
72	C₂×C₇×Q₈	SD₁₆	1	4	yes	—
73	C₂×C₇×Q₈	Q₁₆	1	4	yes	—
74	C₂×C₇×Q₈	C₂²×C₄	1	2	yes	—
75	C₂×C₇×Q₈	C₂×Q₈	1	2	yes	—
76	C₂×C₇×Q₈	#41	1	2	yes	—
77	C₂²×C₄×C₇	—	2	1	no	—
78	C₂²×C₄×C₇	—	2	1	no	—
79	C₂²×C₄×C₇	—	2	3	no	—
80	C₂²×C₄×C₇	—	2	3	no	—
81	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
82	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
83	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
84	C₂×C₇×C₈	—	2	1	no	—
85	C₂×C₇×C₈	—	2	1	no	—
86	C₂×C₇×C₈	—	2	3	no	—
87	C₂×C₇×C₈	—	2	3	no	—
88	C₂×C₇×C₈	—	2	3	no	—
89	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
90	C₇×C₄⋊C₄	—	2	12	no	—
91	C₇×C₄⋊C₄	—	2	12	no	—
92	C₇×C₄⋊C₄	—	2	3	no	—
93	C₇×C₄⋊C₄	—	2	1	no	—
94	C₇×C₄⋊C₄	—	2	3	no	—
95	C₇×C₄⋊C₄	—	2	3	no	—
96	C₇×C₄⋊C₄	—	4	6	no	—
97	C₇×C₄⋊C₄	—	8	3	no	—
98	C₇×C₄⋊C₄	C₂²⋊C₄	1	12	yes	—
99	C₇×C₄⋊C₄	C₂²×C₄	1	6	yes	—
100	C₇×C₄⋊C₄	C₂×Q₈	1	2	yes	—
101	C₄²×C₇	—	2	12	no	—
102	C₄²×C₇	—	4	6	no	—
103	C₄²×C₇	C₂²×C₄	1	6	yes	—
104	C₄²×C₇	C₂×D₄	1	6	yes	—
105	Q₈²	—	2	4	no	—
106	Q₈²	C₄×C₇	1	8	yes	complemented
107	C₂×C₄⋊Q₈	—	2	3	no	—
108	C₂×C₄⋊Q₈	C₂²×C₇	1	3	yes	—
109	C₂×C₄×Q₈	—	2	1	no	—
110	C₂×C₄×Q₈	—	2	1	no	—
111	C₂×C₄×Q₈	C₂²×C₇	1	3	yes	—
112	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
113	C₈×Q₈	—	2	4	no	—
114	C₂×C₄⋊C₈	C₂²×C₇	1	3	yes	—
115	C₂×#103	C₂²×C₇	1	3	yes	—
116	C₂×#103	C₂²×C₇	1	1	yes	characteristic
117	C₂×C₄×C₈	C₂²×C₇	1	3	yes	—
118	#389	—	2	3	no	—
119	#384	C₂²×C₇	1	3	yes	—
120	C₄×C₄⋊C₄	C₂²×C₇	1	3	yes	—
121	#340	C₂²×C₇	1	6	yes	—
122	#340	C₂²×C₇	1	3	yes	—
123	#340	C₂²×C₇	1	3	yes	—
124	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
125	C₇×Q₈	—	2	2	no	—
126	C₇×Q₈	—	2	2	no	—
127	C₇×Q₈	—	2	2	no	—
128	C₇×Q₈	—	8	12	no	—
129	C₇×Q₈	#103	1	16	yes	direct product factor
130	C₇×Q₈	C₄×Q₈	1	4	yes	complemented
131	C₂×C₄×C₇	—	2	2	no	—
132	C₂×C₄×C₇	—	2	2	no	—
133	C₂×C₄×C₇	—	2	2	no	—
134	C₂×C₄×C₇	—	2	3	no	—
135	C₂×C₄×C₇	—	2	3	no	—
136	C₂×C₄×C₇	—	2	6	no	—
137	C₂×C₄×C₇	—	2	2	no	—
138	C₂×C₄×C₇	—	4	1	no	—
139	C₂×C₄×C₇	—	4	3	no	—
140	C₂×C₄×C₇	—	4	1	no	—
141	C₂×C₄×C₇	—	4	3	no	—
142	C₂×C₄×C₇	—	4	3	no	—
143	C₂×C₄×C₇	—	4	3	no	—
144	C₂×C₄×C₇	—	4	3	no	—
145	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	6	yes	—
146	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	6	yes	—
147	C₂×C₄×C₇	C₂×SD₁₆	1	6	yes	—
148	C₂×C₄×C₇	C₂×Q₁₆	1	6	yes	—
149	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
150	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
152	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
153	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
154	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
155	C₇×C₈	—	2	2	no	—
156	C₇×C₈	—	4	3	no	—
157	C₂²×Q₈	—	2	3	no	—
158	C₂²×Q₈	C₇×D₄	1	1	yes	characteristic
159	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
160	C₂²×C₈	C₂³×C₇	1	3	yes	—
161	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
162	C₄⋊Q₈	—	2	24	no	—
163	C₄⋊Q₈	—	2	6	no	—
164	C₄⋊Q₈	C₂×C₄×C₇	1	12	yes	—
165	C₄×Q₈	—	2	8	no	—
166	C₄×Q₈	—	2	6	no	—
167	C₄×Q₈	—	4	4	no	—
168	C₄×Q₈	C₂×C₄×C₇	1	12	yes	—
169	C₄×Q₈	C₂×C₄×C₇	1	4	yes	—
170	C₄×Q₈	C₇×D₄	1	4	yes	—
171	C₂×C₄⋊C₄	—	2	3	no	—
172	C₂×C₄⋊C₄	—	2	3	no	—
173	C₂×C₄⋊C₄	—	2	3	no	—
174	C₂×C₄⋊C₄	—	2	3	no	—
175	C₂×C₄⋊C₄	C₇×D₄	1	3	yes	—
176	C₂×C₄⋊C₄	C₂³×C₇	1	3	yes	—
177	C₂×C₄⋊C₄	C₂³×C₇	1	3	yes	—
178	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
179	C₂×C₄²	—	2	3	no	—
180	C₂×C₄²	—	2	3	no	—
181	C₂×C₄²	C₂³×C₇	1	3	yes	—
182	C₄⋊C₈	—	2	6	no	—
183	#103	—	2	6	no	—
184	#103	C₇×Q₈	1	4	yes	direct product factor
185	C₄×C₈	—	2	6	no	—
186	#95	—	2	3	no	—
187	C₂²×C₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
188	C₂²×C₇	#548	1	1	yes	characteristic
189	C₂²×C₇	D₄×Q₈	1	1	yes	characteristic
190	C₂²×C₇	C₂²×SD₁₆	1	1	yes	characteristic
191	C₂²×C₇	C₂²×Q₁₆	1	1	yes	characteristic
192	C₂²×C₇	#576	1	1	yes	characteristic
193	C₂²×C₇	#577	1	1	yes	characteristic
194	C₄×C₇	—	2	4	no	—
195	C₄×C₇	—	2	6	no	—
196	C₄×C₇	—	4	2	no	—
197	C₄×C₇	—	4	6	no	—
198	C₄×C₇	—	8	3	no	—
199	C₄×C₇	C₂×#103	1	12	yes	—
200	C₄×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
201	C₄×C₇	#519	1	1	yes	characteristic
202	C₄×C₇	#548	1	1	yes	characteristic
203	C₄×C₇	D₄×Q₈	1	1	yes	characteristic
204	C₂×Q₈	—	2	12	no	—
205	C₂×Q₈	—	2	1	no	—
206	C₂×Q₈	—	2	1	no	—
207	C₂×Q₈	—	2	1	no	—
208	C₂×Q₈	—	8	3	no	—
209	C₂×Q₈	—	8	3	no	—
210	C₂×Q₈	C₇×C₂²⋊C₄	1	4	yes	—
211	C₂×Q₈	C₇×SD₁₆	1	4	yes	—
212	C₂×Q₈	C₇×Q₁₆	1	4	yes	—
213	C₂×Q₈	C₂²×C₄×C₇	1	2	yes	—
214	C₂×Q₈	C₂×C₇×Q₈	1	2	yes	—
215	C₂×Q₈	C₇×#41	1	2	yes	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	3	no	—
219	C₂²×C₄	—	2	3	no	—
220	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
221	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
222	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	3	no	—
226	C₂×C₈	—	2	3	no	—
227	C₂×C₈	—	2	3	no	—
228	C₂×C₈	C₂×C₇×Q₈	1	2	yes	—
229	C₄⋊C₄	—	2	12	no	—
230	C₄⋊C₄	—	2	12	no	—
231	C₄⋊C₄	—	2	3	no	—
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	3	no	—
234	C₄⋊C₄	—	2	3	no	—
235	C₄⋊C₄	—	4	6	no	—
236	C₄⋊C₄	—	8	3	no	—
237	C₄⋊C₄	C₇×C₂²⋊C₄	1	12	yes	—
238	C₄⋊C₄	C₂²×C₄×C₇	1	6	yes	—
239	C₄⋊C₄	C₂×C₇×Q₈	1	2	yes	—
240	C₄²	—	2	12	no	—
241	C₄²	—	4	6	no	—
242	C₄²	C₂²×C₄×C₇	1	6	yes	—
243	C₄²	C₂×C₇×D₄	1	6	yes	—
244	C₂×C₇	Q₈×C₂²⋊C₄	1	1	yes	characteristic
245	C₂×C₇	C₂²×#103	1	1	yes	characteristic
246	C₂×C₇	#2897	1	1	yes	characteristic
247	C₂×C₇	#3362	1	1	yes	characteristic
248	C₂×C₇	#3363	1	1	yes	characteristic
249	C₂×C₇	Q₈×SD₁₆	1	1	yes	characteristic
250	C₂×C₇	Q₈×Q₁₆	1	1	yes	characteristic
251	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
252	Q₈	—	2	2	no	—
253	Q₈	—	2	2	no	—
254	Q₈	—	2	2	no	—
255	Q₈	—	8	12	no	—
256	Q₈	C₇×#103	1	16	yes	direct product factor
257	Q₈	C₄×C₇×Q₈	1	4	yes	complemented
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	3	no	—
262	C₂×C₄	—	2	3	no	—
263	C₂×C₄	—	2	6	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	4	3	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	3	no	—
269	C₂×C₄	—	4	3	no	—
270	C₂×C₄	—	4	3	no	—
271	C₂×C₄	—	4	3	no	—
272	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	6	yes	—
273	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	6	yes	—
274	C₂×C₄	C₂×C₇×SD₁₆	1	6	yes	—
275	C₂×C₄	C₂×C₇×Q₁₆	1	6	yes	—
276	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
277	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
278	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
279	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
280	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
281	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
282	C₈	—	2	2	no	—
283	C₈	—	4	3	no	—
284	C₇	Q₈×#103	1	1	yes	characteristic, direct product factor, Sylow
285	C₂²	C₂²×C₇×C₂²⋊C₄	1	1	yes	characteristic
286	C₂²	C₇×#548	1	1	yes	characteristic
287	C₂²	C₇×D₄×Q₈	1	1	yes	characteristic
288	C₂²	C₂²×C₇×SD₁₆	1	1	yes	characteristic
289	C₂²	C₂²×C₇×Q₁₆	1	1	yes	characteristic
290	C₂²	C₇×#576	1	1	yes	characteristic
291	C₂²	C₇×#577	1	1	yes	characteristic
292	C₄	—	2	4	no	—
293	C₄	—	2	6	no	—
294	C₄	—	4	2	no	—
295	C₄	—	4	6	no	—
296	C₄	—	8	3	no	—
297	C₄	C₂×C₇×#103	1	12	yes	—
298	C₄	C₂×C₄×C₇×Q₈	1	1	yes	characteristic
299	C₄	C₇×#519	1	1	yes	characteristic
300	C₄	C₇×#548	1	1	yes	characteristic
301	C₄	C₇×D₄×Q₈	1	1	yes	characteristic
302	C₂	C₇×Q₈×C₂²⋊C₄	1	1	yes	characteristic
303	C₂	C₂²×C₇×#103	1	1	yes	characteristic
304	C₂	C₇×#2897	1	1	yes	characteristic
305	C₂	C₇×#3362	1	1	yes	characteristic
306	C₂	C₇×#3363	1	1	yes	characteristic
307	C₂	C₇×Q₈×SD₁₆	1	1	yes	characteristic
308	C₂	C₇×Q₈×Q₁₆	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄×C₇×Q₈ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₇×Q₈²	C₄	semidirect
27. Q₈×#103	C₇	direct (N × H)
55. C₇×#103	Q₈	direct (N × H)
106. Q₈²	C₄×C₇	semidirect
129. C₇×Q₈	#103	direct (N × H)
130. C₇×Q₈	C₄×Q₈	semidirect
184. #103	C₇×Q₈	direct (N × H)
256. Q₈	C₇×#103	direct (N × H)
257. Q₈	C₄×C₇×Q₈	semidirect
284. C₇	Q₈×#103	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×Q₈²	2
C₂×C₇×C₈×Q₈	2
C₇×Q₈×C₄⋊C₄	2
C₇×#1990	2
C₇×#1987	2
C₇×#1803	2
C₄×C₇×#103	2
Q₈×#103	7

Subgroups — C7×Q8×#103 #13199736

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×Q₈×#103 #13199736