Subgroups — C₂×C₇×#3363 #13238516

Order 1792.

1830 conjugacy classes of subgroups.

The subgroup lattice has 312 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×#557	C₂	1	1	yes	maximal, characteristic
3	C₇×#3363	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₇×#487	C₂	1	3	yes	maximal
5	C₂×C₇×#477	C₂	1	3	yes	maximal
6	C₂×C₇×#462	C₂	1	3	yes	maximal
7	C₂×C₇×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×C₇×Q₁₆	C₂	1	3	yes	maximal
9	C₂²×C₇×Q₁₆	C₂²	1	3	yes	—
10	C₇×Q₈²	—	2	4	no	—
11	C₇×Q₈²	C₂²	1	8	yes	—
12	C₇×#557	—	2	4	no	—
13	C₇×#557	C₂²	1	8	yes	—
14	C₂×C₇×C₄⋊Q₈	—	2	3	no	—
15	C₂×C₇×C₄⋊Q₈	C₂²	1	3	yes	—
16	C₂×C₇×C₄⋊Q₈	C₂²	1	3	yes	—
17	C₂×C₇×#125	—	2	3	no	—
18	C₂×C₄×C₇×Q₈	—	2	1	no	—
19	C₂×C₄×C₇×Q₈	—	2	1	no	—
20	C₂×C₄×C₇×Q₈	C₂²	1	3	yes	—
21	C₂×C₄×C₇×Q₈	C₂²	1	3	yes	—
22	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
23	C₇×#487	C₂²	1	24	yes	—
24	C₇×#477	C₂²	1	24	yes	—
25	C₇×#462	C₂²	1	24	yes	—
26	C₇×C₈×Q₈	C₂²	1	8	yes	—
27	C₄×C₇×Q₁₆	C₂²	1	24	yes	—
28	C₂×C₇×#107	C₂²	1	3	yes	—
29	C₂×C₇×C₄⋊C₈	C₂²	1	3	yes	—
30	C₂×C₇×#103	C₂²	1	1	yes	characteristic
31	C₂×C₇×#103	C₂²	1	3	yes	—
32	C₂×C₇×#103	C₂²	1	6	yes	—
33	C₂×C₄×C₇×C₈	C₂²	1	3	yes	—
34	C₂×#3363	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	C₂²×C₇×Q₈	—	2	3	no	—
36	C₂²×C₇×Q₈	D₄	1	1	yes	characteristic
37	C₂²×C₇×Q₈	C₂³	1	3	yes	—
38	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
39	C₂×C₇×Q₁₆	—	2	12	no	—
40	C₂×C₇×Q₁₆	C₂³	1	12	yes	—
41	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₇×C₈	C₂³	1	3	yes	—
43	C₇×C₄⋊Q₈	—	2	24	no	—
44	C₇×C₄⋊Q₈	—	2	6	no	—
45	C₇×C₄⋊Q₈	—	2	6	no	—
46	C₇×C₄⋊Q₈	C₂³	1	12	yes	—
47	C₇×C₄⋊Q₈	C₂³	1	12	yes	—
48	C₇×#125	—	2	24	no	—
49	C₄×C₇×Q₈	—	2	8	no	—
50	C₄×C₇×Q₈	—	2	8	no	—
51	C₄×C₇×Q₈	—	2	6	no	—
52	C₄×C₇×Q₈	—	2	6	no	—
53	C₄×C₇×Q₈	D₄	1	4	yes	—
54	C₄×C₇×Q₈	C₂³	1	12	yes	—
55	C₄×C₇×Q₈	C₂³	1	12	yes	—
56	C₄×C₇×Q₈	C₂³	1	4	yes	—
57	C₂×C₇×C₄⋊C₄	—	2	3	no	—
58	C₂×C₇×C₄⋊C₄	—	2	3	no	—
59	C₂×C₇×C₄⋊C₄	—	2	3	no	—
60	C₂×C₇×C₄⋊C₄	—	2	3	no	—
61	C₂×C₇×C₄⋊C₄	—	2	3	no	—
62	C₂×C₇×C₄⋊C₄	D₄	1	3	yes	—
63	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
64	C₂×C₇×C₄⋊C₄	C₂³	1	3	yes	—
65	C₂×C₄²×C₇	—	2	3	no	—
66	C₂×C₄²×C₇	—	2	3	no	—
67	C₂×C₄²×C₇	C₂³	1	3	yes	—
68	C₇×#107	C₂³	1	12	yes	—
69	C₇×C₄⋊C₈	C₂³	1	12	yes	—
70	C₇×#103	C₂³	1	24	yes	—
71	C₇×#103	C₂³	1	12	yes	—
72	C₇×#103	C₂³	1	4	yes	—
73	C₄×C₇×C₈	C₂³	1	12	yes	—
74	C₂×Q₈²	C₂×C₇	1	1	yes	characteristic
75	C₂×#557	C₂×C₇	1	1	yes	characteristic
76	#3363	C₂×C₇	1	16	yes	direct product factor
77	C₂×#487	C₂×C₇	1	3	yes	—
78	C₂×#477	C₂×C₇	1	3	yes	—
79	C₂×#462	C₂×C₇	1	3	yes	—
80	C₂×C₈×Q₈	C₂×C₇	1	1	yes	characteristic
81	C₂×C₄×Q₁₆	C₂×C₇	1	3	yes	—
82	C₂×C₇×Q₈	—	2	1	no	—
83	C₂×C₇×Q₈	—	2	3	no	—
84	C₂×C₇×Q₈	—	2	12	no	—
85	C₂×C₇×Q₈	—	2	2	no	—
86	C₂×C₇×Q₈	—	2	6	no	—
87	C₂×C₇×Q₈	—	2	6	no	—
88	C₂×C₇×Q₈	—	8	6	no	—
89	C₂×C₇×Q₈	Q₁₆	1	8	yes	complemented
90	C₂×C₇×Q₈	C₂×D₄	1	4	yes	—
91	C₂×C₇×Q₈	#41	1	4	yes	—
92	C₂×C₇×Q₈	C₂⁴	1	6	yes	—
93	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
94	C₂²×C₄×C₇	—	2	1	no	—
95	C₂²×C₄×C₇	—	2	1	no	—
96	C₂²×C₄×C₇	—	2	3	no	—
97	C₂²×C₄×C₇	—	2	3	no	—
98	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
99	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
100	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
101	C₇×Q₁₆	—	2	24	no	—
102	C₂×C₇×C₈	—	2	6	no	—
103	C₂×C₇×C₈	—	2	2	no	—
104	C₂×C₇×C₈	C₂⁴	1	2	yes	—
105	C₂×C₇×C₈	C₂⁴	1	6	yes	—
106	C₇×C₄⋊C₄	—	2	12	no	—
107	C₇×C₄⋊C₄	—	2	12	no	—
108	C₇×C₄⋊C₄	—	2	12	no	—
109	C₇×C₄⋊C₄	—	2	12	no	—
110	C₇×C₄⋊C₄	—	2	12	no	—
111	C₇×C₄⋊C₄	—	2	1	no	—
112	C₇×C₄⋊C₄	—	2	3	no	—
113	C₇×C₄⋊C₄	C₂×D₄	1	12	yes	—
114	C₇×C₄⋊C₄	C₂⁴	1	6	yes	—
115	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
116	C₄²×C₇	—	2	12	no	—
117	C₄²×C₇	—	2	12	no	—
118	C₄²×C₇	C₂×D₄	1	6	yes	—
119	C₄²×C₇	C₂⁴	1	6	yes	—
120	C₂²×Q₁₆	C₂²×C₇	1	3	yes	—
121	Q₈²	—	2	4	no	—
122	Q₈²	C₂²×C₇	1	8	yes	—
123	#557	—	2	4	no	—
124	#557	C₂²×C₇	1	8	yes	—
125	C₂×C₄⋊Q₈	—	2	3	no	—
126	C₂×C₄⋊Q₈	C₂²×C₇	1	3	yes	—
127	C₂×C₄⋊Q₈	C₂²×C₇	1	3	yes	—
128	C₂×#125	—	2	3	no	—
129	C₂×C₄×Q₈	—	2	1	no	—
130	C₂×C₄×Q₈	—	2	1	no	—
131	C₂×C₄×Q₈	C₂²×C₇	1	3	yes	—
132	C₂×C₄×Q₈	C₂²×C₇	1	3	yes	—
133	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
134	#487	C₂²×C₇	1	24	yes	—
135	#477	C₂²×C₇	1	24	yes	—
136	#462	C₂²×C₇	1	24	yes	—
137	C₈×Q₈	C₂²×C₇	1	8	yes	—
138	C₄×Q₁₆	C₂²×C₇	1	24	yes	—
139	C₂×#107	C₂²×C₇	1	3	yes	—
140	C₂×C₄⋊C₈	C₂²×C₇	1	3	yes	—
141	C₂×#103	C₂²×C₇	1	6	yes	—
142	C₂×#103	C₂²×C₇	1	3	yes	—
143	C₂×#103	C₂²×C₇	1	1	yes	characteristic
144	C₂×C₄×C₈	C₂²×C₇	1	3	yes	—
145	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
146	C₇×Q₈	—	2	4	no	—
147	C₇×Q₈	—	2	2	no	—
148	C₇×Q₈	—	2	6	no	—
149	C₇×Q₈	—	2	6	no	—
150	C₇×Q₈	—	4	6	no	—
151	C₇×Q₈	—	8	12	no	—
152	C₇×Q₈	C₂×Q₁₆	1	16	yes	complemented
153	C₇×Q₈	C₂×#41	1	4	yes	—
154	C₂×C₄×C₇	—	2	4	no	—
155	C₂×C₄×C₇	—	2	6	no	—
156	C₂×C₄×C₇	—	2	2	no	—
157	C₂×C₄×C₇	—	2	2	no	—
158	C₂×C₄×C₇	—	2	6	no	—
159	C₂×C₄×C₇	—	2	6	no	—
160	C₂×C₄×C₇	—	4	6	no	—
161	C₂×C₄×C₇	—	4	2	no	—
162	C₂×C₄×C₇	—	4	6	no	—
163	C₂×C₄×C₇	C₂×Q₁₆	1	12	yes	—
164	C₂×C₄×C₇	C₂²×D₄	1	6	yes	—
165	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
166	C₂×C₄×C₇	C₂²×D₄	1	6	yes	—
167	C₂×C₄×C₇	C₂×#41	1	2	yes	—
168	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
169	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
170	C₇×C₈	—	2	6	no	—
171	C₇×C₈	—	2	2	no	—
172	C₂²×Q₈	—	2	3	no	—
173	C₂²×Q₈	C₇×D₄	1	1	yes	characteristic
174	C₂²×Q₈	C₂³×C₇	1	3	yes	—
175	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
176	C₂×Q₁₆	—	2	12	no	—
177	C₂×Q₁₆	C₂³×C₇	1	12	yes	—
178	C₂²×C₈	C₂³×C₇	1	3	yes	—
179	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
180	C₄⋊Q₈	—	2	24	no	—
181	C₄⋊Q₈	—	2	6	no	—
182	C₄⋊Q₈	—	2	6	no	—
183	C₄⋊Q₈	C₂³×C₇	1	12	yes	—
184	C₄⋊Q₈	C₂³×C₇	1	12	yes	—
185	#125	—	2	24	no	—
186	C₄×Q₈	—	2	8	no	—
187	C₄×Q₈	—	2	8	no	—
188	C₄×Q₈	—	2	6	no	—
189	C₄×Q₈	—	2	6	no	—
190	C₄×Q₈	C₇×D₄	1	4	yes	—
191	C₄×Q₈	C₂³×C₇	1	12	yes	—
192	C₄×Q₈	C₂³×C₇	1	12	yes	—
193	C₄×Q₈	C₂³×C₇	1	4	yes	—
194	C₂×C₄⋊C₄	—	2	3	no	—
195	C₂×C₄⋊C₄	—	2	3	no	—
196	C₂×C₄⋊C₄	—	2	3	no	—
197	C₂×C₄⋊C₄	—	2	3	no	—
198	C₂×C₄⋊C₄	—	2	3	no	—
199	C₂×C₄⋊C₄	C₇×D₄	1	3	yes	—
200	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
201	C₂×C₄⋊C₄	C₂³×C₇	1	3	yes	—
202	C₂×C₄²	—	2	3	no	—
203	C₂×C₄²	—	2	3	no	—
204	C₂×C₄²	C₂³×C₇	1	3	yes	—
205	#107	C₂³×C₇	1	12	yes	—
206	C₄⋊C₈	C₂³×C₇	1	12	yes	—
207	#103	C₂³×C₇	1	4	yes	—
208	#103	C₂³×C₇	1	12	yes	—
209	#103	C₂³×C₇	1	24	yes	—
210	C₄×C₈	C₂³×C₇	1	12	yes	—
211	C₂²×C₇	#548	1	2	yes	—
212	C₂²×C₇	C₂²×Q₁₆	1	2	yes	—
213	C₂²×C₇	#577	1	2	yes	—
214	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
215	C₄×C₇	—	2	4	no	—
216	C₄×C₇	—	2	6	no	—
217	C₄×C₇	—	4	6	no	—
218	C₄×C₇	—	4	2	no	—
219	C₄×C₇	—	4	6	no	—
220	C₄×C₇	#548	1	2	yes	—
221	C₄×C₇	C₂²×Q₁₆	1	12	yes	—
222	C₄×C₇	C₂²×#41	1	1	yes	characteristic
223	C₄×C₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
224	C₂×Q₈	—	2	1	no	—
225	C₂×Q₈	—	2	3	no	—
226	C₂×Q₈	—	2	12	no	—
227	C₂×Q₈	—	2	2	no	—
228	C₂×Q₈	—	2	6	no	—
229	C₂×Q₈	—	2	6	no	—
230	C₂×Q₈	—	8	6	no	—
231	C₂×Q₈	C₇×Q₁₆	1	8	yes	complemented
232	C₂×Q₈	C₂×C₇×D₄	1	4	yes	—
233	C₂×Q₈	C₇×#41	1	4	yes	—
234	C₂×Q₈	C₂⁴×C₇	1	6	yes	—
235	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	2	3	no	—
239	C₂²×C₄	—	2	3	no	—
240	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
241	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
242	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
243	Q₁₆	—	2	24	no	—
244	C₂×C₈	—	2	6	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	C₂⁴×C₇	1	2	yes	—
247	C₂×C₈	C₂⁴×C₇	1	6	yes	—
248	C₄⋊C₄	—	2	12	no	—
249	C₄⋊C₄	—	2	12	no	—
250	C₄⋊C₄	—	2	12	no	—
251	C₄⋊C₄	—	2	12	no	—
252	C₄⋊C₄	—	2	12	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	3	no	—
255	C₄⋊C₄	C₂×C₇×D₄	1	12	yes	—
256	C₄⋊C₄	C₂⁴×C₇	1	6	yes	—
257	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
258	C₄²	—	2	12	no	—
259	C₄²	—	2	12	no	—
260	C₄²	C₂×C₇×D₄	1	6	yes	—
261	C₄²	C₂⁴×C₇	1	6	yes	—
262	C₂×C₇	#3363	1	4	yes	direct product factor
263	C₂×C₇	C₂×#548	1	1	yes	characteristic
264	C₂×C₇	C₂³×Q₁₆	1	1	yes	characteristic
265	C₂×C₇	C₂×#577	1	1	yes	characteristic
266	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
267	Q₈	—	2	2	no	—
268	Q₈	—	2	4	no	—
269	Q₈	—	2	6	no	—
270	Q₈	—	2	6	no	—
271	Q₈	—	4	6	no	—
272	Q₈	—	8	12	no	—
273	Q₈	C₂×C₇×Q₁₆	1	16	yes	complemented
274	Q₈	C₂×C₇×#41	1	4	yes	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	6	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	6	no	—
280	C₂×C₄	—	2	6	no	—
281	C₂×C₄	—	4	6	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	6	no	—
284	C₂×C₄	C₂×C₇×Q₁₆	1	12	yes	—
285	C₂×C₄	C₂²×C₇×D₄	1	6	yes	—
286	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
287	C₂×C₄	C₂²×C₇×D₄	1	6	yes	—
288	C₂×C₄	C₂×C₇×#41	1	2	yes	—
289	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
290	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
291	C₈	—	2	6	no	—
292	C₈	—	2	2	no	—
293	C₇	C₂×#3363	1	1	yes	characteristic, direct product factor, Sylow
294	C₂²	C₇×#548	1	2	yes	—
295	C₂²	C₂²×C₇×Q₁₆	1	2	yes	—
296	C₂²	C₇×#577	1	2	yes	—
297	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
298	C₄	—	2	6	no	—
299	C₄	—	2	4	no	—
300	C₄	—	4	6	no	—
301	C₄	—	4	2	no	—
302	C₄	—	4	6	no	—
303	C₄	C₇×#548	1	2	yes	—
304	C₄	C₂²×C₇×Q₁₆	1	12	yes	—
305	C₄	C₂²×C₇×#41	1	1	yes	characteristic
306	C₄	C₂×C₇×2₋¹⁺⁴	1	1	yes	characteristic
307	C₂	C₇×#3363	1	4	yes	direct product factor
308	C₂	C₂×C₇×#548	1	1	yes	characteristic
309	C₂	C₂³×C₇×Q₁₆	1	1	yes	characteristic
310	C₂	C₂×C₇×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×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
3. C₇×#3363	C₂	direct (N × H)
34. C₂×#3363	C₇	direct (N × H)
76. #3363	C₂×C₇	direct (N × H)
89. C₂×C₇×Q₈	Q₁₆	semidirect
152. C₇×Q₈	C₂×Q₁₆	semidirect
231. C₂×Q₈	C₇×Q₁₆	semidirect
262. C₂×C₇	#3363	direct (N × H)
273. Q₈	C₂×C₇×Q₁₆	semidirect
293. C₇	C₂×#3363	direct (N × H)
307. C₂	C₇×#3363	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×#557	2
C₇×#3363	2
C₂×C₇×#487	2
C₂×C₇×#477	2
C₂×C₇×#462	2
C₂×C₇×C₈×Q₈	2
C₂×C₄×C₇×Q₁₆	2
C₂×#3363	7

Subgroups — C2×C7×#3363 #13238516

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#3363 #13238516