Subgroups — C₇×Q₈×#102 #13199734

Order 1792.

1314 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×D₄×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×C₈×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×Q₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#1990	C₂	1	3	yes	maximal, complemented
5	C₇×#1988	C₂	1	3	yes	maximal, complemented
6	C₇×#1802	C₂	1	3	yes	maximal
7	C₄×C₇×#102	C₂	1	3	yes	maximal
8	C₂³×C₇×Q₈	—	2	1	no	—
9	C₇×D₄×Q₈	—	2	4	no	—
10	C₇×D₄×Q₈	C₄	1	8	yes	complemented
11	C₂×C₇×C₄⋊Q₈	C₂²	1	3	yes	—
12	C₂×C₇×C₂²⋊Q₈	—	2	3	no	—
13	C₂×C₄×C₇×Q₈	—	2	1	no	—
14	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₇×D₄	C₂²	1	3	yes	—
16	C₇×C₈×Q₈	—	2	4	no	—
17	C₂×C₇×C₄⋊C₈	C₂²	1	3	yes	—
18	C₂×C₇×#102	C₂²	1	3	yes	—
19	C₂×C₇×#102	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×#102	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂²×C₇×Q₈	—	2	8	no	—
29	C₂²×C₇×Q₈	—	2	3	no	—
30	C₂²×C₇×Q₈	—	4	4	no	—
31	C₂²×C₇×Q₈	—	4	4	no	—
32	C₂²×C₇×Q₈	D₄	1	1	yes	characteristic
33	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
34	C₂³×C₄×C₇	—	2	3	no	—
35	C₂²×C₇×C₈	C₂³	1	3	yes	—
36	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
37	C₇×C₄⋊Q₈	—	2	6	no	—
38	C₇×C₄⋊Q₈	C₂×C₄	1	12	yes	—
39	C₇×C₂²⋊Q₈	—	2	24	no	—
40	C₄×C₇×Q₈	—	4	4	no	—
41	C₄×C₇×Q₈	C₂×C₄	1	4	yes	—
42	C₄×C₇×Q₈	D₄	1	4	yes	—
43	C₄×C₇×D₄	—	2	6	no	—
44	C₄×C₇×D₄	C₂×C₄	1	12	yes	—
45	C₂×C₇×C₄⋊C₄	—	2	3	no	—
46	C₂×C₇×C₄⋊C₄	—	2	3	no	—
47	C₂×C₇×C₄⋊C₄	—	2	3	no	—
48	C₂×C₇×C₄⋊C₄	D₄	1	3	yes	—
49	C₂×C₇×C₄⋊C₄	C₂³	1	3	yes	—
50	C₂×C₇×C₄⋊C₄	C₂³	1	3	yes	—
51	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
52	C₂×C₇×C₂²⋊C₄	—	2	3	no	—
53	C₂×C₄²×C₇	—	2	3	no	—
54	C₂×C₄²×C₇	C₂³	1	3	yes	—
55	C₇×C₄⋊C₈	—	2	6	no	—
56	C₇×#102	—	2	6	no	—
57	C₇×#102	Q₈	1	4	yes	direct product factor
58	C₄×C₇×C₈	—	2	6	no	—
59	C₇×#95	—	2	3	no	—
60	C₂×D₄×Q₈	C₂×C₇	1	1	yes	characteristic
61	C₂×C₈×Q₈	C₂×C₇	1	1	yes	characteristic, complemented
62	Q₈×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
63	#1990	C₂×C₇	1	3	yes	complemented
64	#1988	C₂×C₇	1	3	yes	complemented
65	#1802	C₂×C₇	1	3	yes	—
66	C₄×#102	C₂×C₇	1	3	yes	—
67	C₂⁴×C₇	—	2	1	no	—
68	C₂×C₇×Q₈	—	2	12	no	—
69	C₂×C₇×Q₈	—	4	32	no	—
70	C₂×C₇×Q₈	—	4	6	no	—
71	C₂×C₇×Q₈	—	4	6	no	—
72	C₂×C₇×Q₈	C₂²⋊C₄	1	4	yes	—
73	C₂×C₇×Q₈	D₈	1	4	yes	—
74	C₂×C₇×Q₈	SD₁₆	1	4	yes	—
75	C₂×C₇×D₄	—	2	1	no	—
76	C₂×C₇×D₄	—	2	1	no	—
77	C₂×C₇×D₄	—	2	1	no	—
78	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
79	C₂×C₇×D₄	C₂×Q₈	1	2	yes	—
80	C₂×C₇×D₄	#41	1	2	yes	—
81	C₂²×C₄×C₇	—	2	1	no	—
82	C₂²×C₄×C₇	—	2	3	no	—
83	C₂²×C₄×C₇	—	2	12	no	—
84	C₂²×C₄×C₇	—	2	3	no	—
85	C₂²×C₄×C₇	—	4	6	no	—
86	C₂²×C₄×C₇	—	4	6	no	—
87	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
88	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
89	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
90	C₂×C₇×C₈	—	2	1	no	—
91	C₂×C₇×C₈	—	2	1	no	—
92	C₂×C₇×C₈	—	2	3	no	—
93	C₂×C₇×C₈	—	2	3	no	—
94	C₂×C₇×C₈	—	2	3	no	—
95	C₂×C₇×C₈	C₂×Q₈	1	2	yes	complemented
96	C₇×C₄⋊C₄	—	2	12	no	—
97	C₇×C₄⋊C₄	—	2	3	no	—
98	C₇×C₄⋊C₄	—	2	1	no	—
99	C₇×C₄⋊C₄	—	2	3	no	—
100	C₇×C₄⋊C₄	—	2	3	no	—
101	C₇×C₄⋊C₄	—	4	6	no	—
102	C₇×C₄⋊C₄	—	8	3	no	—
103	C₇×C₄⋊C₄	C₂²⋊C₄	1	12	yes	—
104	C₇×C₄⋊C₄	C₂²×C₄	1	6	yes	—
105	C₇×C₄⋊C₄	C₂×Q₈	1	2	yes	complemented
106	C₇×C₂²⋊C₄	—	2	12	no	—
107	C₄²×C₇	—	4	6	no	—
108	C₄²×C₇	C₂²×C₄	1	6	yes	—
109	C₄²×C₇	C₂×D₄	1	6	yes	—
110	C₂³×Q₈	—	2	1	no	—
111	D₄×Q₈	—	2	4	no	—
112	D₄×Q₈	C₄×C₇	1	8	yes	complemented
113	C₂×C₄⋊Q₈	C₂²×C₇	1	3	yes	—
114	C₂×C₂²⋊Q₈	—	2	3	no	—
115	C₂×C₄×Q₈	—	2	1	no	—
116	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
117	C₂×C₄×D₄	C₂²×C₇	1	3	yes	—
118	C₈×Q₈	—	2	4	no	—
119	C₂×C₄⋊C₈	C₂²×C₇	1	3	yes	—
120	C₂×#102	C₂²×C₇	1	3	yes	—
121	C₂×#102	C₂²×C₇	1	1	yes	characteristic
122	C₂×C₄×C₈	C₂²×C₇	1	3	yes	—
123	#389	—	2	3	no	—
124	#384	C₂²×C₇	1	3	yes	—
125	C₄×C₄⋊C₄	C₂²×C₇	1	3	yes	—
126	#340	C₂²×C₇	1	6	yes	—
127	#340	C₂²×C₇	1	3	yes	—
128	#340	C₂²×C₇	1	3	yes	—
129	C₂³×C₇	—	2	2	no	—
130	C₂³×C₇	—	2	2	no	—
131	C₂³×C₇	—	2	2	no	—
132	C₂³×C₇	—	4	1	no	—
133	C₂³×C₇	—	4	1	no	—
134	C₂³×C₇	—	4	1	no	—
135	C₂³×C₇	—	4	1	no	—
136	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
137	C₇×Q₈	—	4	24	no	—
138	C₇×Q₈	#102	1	16	yes	direct product factor
139	C₇×D₄	—	2	2	no	—
140	C₇×D₄	—	2	2	no	—
141	C₇×D₄	—	2	2	no	—
142	C₇×D₄	C₄×Q₈	1	4	yes	complemented
143	C₂×C₄×C₇	—	2	3	no	—
144	C₂×C₄×C₇	—	2	3	no	—
145	C₂×C₄×C₇	—	2	6	no	—
146	C₂×C₄×C₇	—	2	2	no	—
147	C₂×C₄×C₇	—	4	1	no	—
148	C₂×C₄×C₇	—	4	3	no	—
149	C₂×C₄×C₇	—	4	3	no	—
150	C₂×C₄×C₇	—	4	1	no	—
151	C₂×C₄×C₇	—	4	24	no	—
152	C₂×C₄×C₇	—	4	3	no	—
153	C₂×C₄×C₇	—	4	3	no	—
154	C₂×C₄×C₇	—	4	3	no	—
155	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	6	yes	—
156	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	6	yes	—
157	C₂×C₄×C₇	C₂×D₈	1	6	yes	—
158	C₂×C₄×C₇	C₂×SD₁₆	1	6	yes	—
159	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
160	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
162	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
163	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
164	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
165	C₇×C₈	—	2	2	no	—
166	C₇×C₈	—	4	3	no	—
167	C₂²×Q₈	—	2	8	no	—
168	C₂²×Q₈	—	2	3	no	—
169	C₂²×Q₈	—	4	4	no	—
170	C₂²×Q₈	—	4	4	no	—
171	C₂²×Q₈	C₇×D₄	1	1	yes	characteristic
172	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
173	C₂³×C₄	—	2	3	no	—
174	C₂²×C₈	C₂³×C₇	1	3	yes	—
175	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
176	C₄⋊Q₈	—	2	6	no	—
177	C₄⋊Q₈	C₂×C₄×C₇	1	12	yes	—
178	C₂²⋊Q₈	—	2	24	no	—
179	C₄×Q₈	—	4	4	no	—
180	C₄×Q₈	C₂×C₄×C₇	1	4	yes	—
181	C₄×Q₈	C₇×D₄	1	4	yes	—
182	C₄×D₄	—	2	6	no	—
183	C₄×D₄	C₂×C₄×C₇	1	12	yes	—
184	C₂×C₄⋊C₄	—	2	3	no	—
185	C₂×C₄⋊C₄	—	2	3	no	—
186	C₂×C₄⋊C₄	—	2	3	no	—
187	C₂×C₄⋊C₄	C₇×D₄	1	3	yes	—
188	C₂×C₄⋊C₄	C₂³×C₇	1	3	yes	—
189	C₂×C₄⋊C₄	C₂³×C₇	1	3	yes	—
190	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
191	C₂×C₂²⋊C₄	—	2	3	no	—
192	C₂×C₄²	—	2	3	no	—
193	C₂×C₄²	C₂³×C₇	1	3	yes	—
194	C₄⋊C₈	—	2	6	no	—
195	#102	—	2	6	no	—
196	#102	C₇×Q₈	1	4	yes	direct product factor
197	C₄×C₈	—	2	6	no	—
198	#95	—	2	3	no	—
199	C₂²×C₇	—	2	4	no	—
200	C₂²×C₇	—	4	2	no	—
201	C₂²×C₇	—	4	2	no	—
202	C₂²×C₇	—	4	2	no	—
203	C₂²×C₇	—	4	2	no	—
204	C₂²×C₇	—	4	2	no	—
205	C₂²×C₇	—	4	2	no	—
206	C₂²×C₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
207	C₂²×C₇	#548	1	1	yes	characteristic
208	C₂²×C₇	D₄×Q₈	1	1	yes	characteristic
209	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
210	C₂²×C₇	C₂²×SD₁₆	1	1	yes	characteristic
211	C₂²×C₇	#577	1	1	yes	characteristic
212	C₂²×C₇	#578	1	1	yes	characteristic
213	C₄×C₇	—	2	6	no	—
214	C₄×C₇	—	4	2	no	—
215	C₄×C₇	—	4	6	no	—
216	C₄×C₇	—	8	3	no	—
217	C₄×C₇	C₂×#102	1	12	yes	—
218	C₄×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
219	C₄×C₇	#519	1	1	yes	characteristic
220	C₄×C₇	#548	1	1	yes	characteristic
221	C₄×C₇	D₄×Q₈	1	1	yes	characteristic
222	C₂⁴	—	2	1	no	—
223	C₂×Q₈	—	2	12	no	—
224	C₂×Q₈	—	4	32	no	—
225	C₂×Q₈	—	4	6	no	—
226	C₂×Q₈	—	4	6	no	—
227	C₂×Q₈	C₇×C₂²⋊C₄	1	4	yes	—
228	C₂×Q₈	C₇×D₈	1	4	yes	—
229	C₂×Q₈	C₇×SD₁₆	1	4	yes	—
230	C₂×D₄	—	2	1	no	—
231	C₂×D₄	—	2	1	no	—
232	C₂×D₄	—	2	1	no	—
233	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
234	C₂×D₄	C₂×C₇×Q₈	1	2	yes	—
235	C₂×D₄	C₇×#41	1	2	yes	—
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	2	3	no	—
238	C₂²×C₄	—	2	12	no	—
239	C₂²×C₄	—	2	3	no	—
240	C₂²×C₄	—	4	6	no	—
241	C₂²×C₄	—	4	6	no	—
242	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
243	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
244	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	3	no	—
248	C₂×C₈	—	2	3	no	—
249	C₂×C₈	—	2	3	no	—
250	C₂×C₈	C₂×C₇×Q₈	1	2	yes	complemented
251	C₄⋊C₄	—	2	12	no	—
252	C₄⋊C₄	—	2	3	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	3	no	—
255	C₄⋊C₄	—	2	3	no	—
256	C₄⋊C₄	—	4	6	no	—
257	C₄⋊C₄	—	8	3	no	—
258	C₄⋊C₄	C₇×C₂²⋊C₄	1	12	yes	—
259	C₄⋊C₄	C₂²×C₄×C₇	1	6	yes	—
260	C₄⋊C₄	C₂×C₇×Q₈	1	2	yes	complemented
261	C₂²⋊C₄	—	2	12	no	—
262	C₄²	—	4	6	no	—
263	C₄²	C₂²×C₄×C₇	1	6	yes	—
264	C₄²	C₂×C₇×D₄	1	6	yes	—
265	C₂×C₇	—	4	4	no	—
266	C₂×C₇	Q₈×C₂²⋊C₄	1	1	yes	characteristic
267	C₂×C₇	C₂²×#102	1	1	yes	characteristic
268	C₂×C₇	#2898	1	1	yes	characteristic
269	C₂×C₇	#3358	1	1	yes	characteristic
270	C₂×C₇	#3359	1	1	yes	characteristic
271	C₂×C₇	Q₈×D₈	1	1	yes	characteristic
272	C₂×C₇	Q₈×SD₁₆	1	1	yes	characteristic
273	C₂³	—	2	2	no	—
274	C₂³	—	2	2	no	—
275	C₂³	—	2	2	no	—
276	C₂³	—	4	1	no	—
277	C₂³	—	4	1	no	—
278	C₂³	—	4	1	no	—
279	C₂³	—	4	1	no	—
280	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
281	Q₈	—	4	24	no	—
282	Q₈	C₇×#102	1	16	yes	direct product factor
283	D₄	—	2	2	no	—
284	D₄	—	2	2	no	—
285	D₄	—	2	2	no	—
286	D₄	C₄×C₇×Q₈	1	4	yes	complemented
287	C₂×C₄	—	2	3	no	—
288	C₂×C₄	—	2	3	no	—
289	C₂×C₄	—	2	6	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	3	no	—
293	C₂×C₄	—	4	3	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	24	no	—
296	C₂×C₄	—	4	3	no	—
297	C₂×C₄	—	4	3	no	—
298	C₂×C₄	—	4	3	no	—
299	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	6	yes	—
300	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	6	yes	—
301	C₂×C₄	C₂×C₇×D₈	1	6	yes	—
302	C₂×C₄	C₂×C₇×SD₁₆	1	6	yes	—
303	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
304	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
305	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
306	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
307	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
308	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
309	C₈	—	2	2	no	—
310	C₈	—	4	3	no	—
311	C₇	Q₈×#102	1	1	yes	characteristic, direct product factor, Sylow
312	C₂²	—	2	4	no	—
313	C₂²	—	4	2	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	4	2	no	—
318	C₂²	—	4	2	no	—
319	C₂²	C₂²×C₇×C₂²⋊C₄	1	1	yes	characteristic
320	C₂²	C₇×#548	1	1	yes	characteristic
321	C₂²	C₇×D₄×Q₈	1	1	yes	characteristic
322	C₂²	C₂²×C₇×D₈	1	1	yes	characteristic
323	C₂²	C₂²×C₇×SD₁₆	1	1	yes	characteristic
324	C₂²	C₇×#577	1	1	yes	characteristic
325	C₂²	C₇×#578	1	1	yes	characteristic
326	C₄	—	2	6	no	—
327	C₄	—	4	2	no	—
328	C₄	—	4	6	no	—
329	C₄	—	8	3	no	—
330	C₄	C₂×C₇×#102	1	12	yes	—
331	C₄	C₂×C₄×C₇×Q₈	1	1	yes	characteristic
332	C₄	C₇×#519	1	1	yes	characteristic
333	C₄	C₇×#548	1	1	yes	characteristic
334	C₄	C₇×D₄×Q₈	1	1	yes	characteristic
335	C₂	—	4	4	no	—
336	C₂	C₇×Q₈×C₂²⋊C₄	1	1	yes	characteristic
337	C₂	C₂²×C₇×#102	1	1	yes	characteristic
338	C₂	C₇×#2898	1	1	yes	characteristic
339	C₂	C₇×#3358	1	1	yes	characteristic
340	C₂	C₇×#3359	1	1	yes	characteristic
341	C₂	C₇×Q₈×D₈	1	1	yes	characteristic
342	C₂	C₇×Q₈×SD₁₆	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₂²×C₇×D₄	C₂³
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
2. C₂×C₇×C₈×Q₈	C₂	semidirect
3. C₇×Q₈×C₄⋊C₄	C₂	semidirect
4. C₇×#1990	C₂	semidirect
5. C₇×#1988	C₂	semidirect
10. C₇×D₄×Q₈	C₄	semidirect
27. Q₈×#102	C₇	direct (N × H)
57. C₇×#102	Q₈	direct (N × H)
61. C₂×C₈×Q₈	C₂×C₇	semidirect
62. Q₈×C₄⋊C₄	C₂×C₇	semidirect
63. #1990	C₂×C₇	semidirect
64. #1988	C₂×C₇	semidirect
95. C₂×C₇×C₈	C₂×Q₈	semidirect
105. C₇×C₄⋊C₄	C₂×Q₈	semidirect
112. D₄×Q₈	C₄×C₇	semidirect
138. C₇×Q₈	#102	direct (N × H)
142. C₇×D₄	C₄×Q₈	semidirect
196. #102	C₇×Q₈	direct (N × H)
250. C₂×C₈	C₂×C₇×Q₈	semidirect
260. C₄⋊C₄	C₂×C₇×Q₈	semidirect
282. Q₈	C₇×#102	direct (N × H)
286. D₄	C₄×C₇×Q₈	semidirect
311. C₇	Q₈×#102	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×D₄×Q₈	2
C₂×C₇×C₈×Q₈	2
C₇×Q₈×C₄⋊C₄	2
C₇×#1990	2
C₇×#1988	2
C₇×#1802	2
C₄×C₇×#102	2
Q₈×#102	7

Subgroups — C7×Q8×#102 #13199734

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×Q₈×#102 #13199734