Subgroups — C₂×#11725618 #13268654

Order 1792.

630 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×#6389	C₂	1	1	yes	maximal, characteristic
3	C₂²×#6386	C₂	1	1	yes	maximal, characteristic
4	#11725618	C₂	1	4	yes	maximal, direct product factor
5	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
6	C₂²×C₄×Dic₇	C₄	1	1	yes	characteristic
7	C₂³×C₇⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₇×M₃₂	—	2	2	no	—
9	C₂×C₇×M₃₂	—	2	1	no	—
10	C₂×C₇×M₃₂	C₄	1	4	yes	complemented
11	C₂×C₇×M₃₂	C₂²	1	2	yes	—
12	C₂²×C₇×C₁₆	—	2	1	no	—
13	C₂×#6389	—	2	1	no	—
14	C₂×#6389	—	2	1	no	—
15	C₂×#6389	—	2	1	no	—
16	C₂×#6389	—	2	1	no	—
17	C₂×#6389	C₄	1	2	yes	—
18	C₂×#6389	C₂²	1	2	yes	—
19	C₂×#6386	—	2	2	no	—
20	C₂×#6386	—	2	1	no	—
21	C₂×#6386	C₄	1	4	yes	complemented
22	C₂×#6386	C₂²	1	2	yes	—
23	C₂²×C₇⋊C₁₆	—	2	1	no	—
24	C₂×#1377	—	7	1	no	maximal, Sylow
25	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
26	C₂³×Dic₇	C₈	1	2	yes	—
27	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
29	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
30	C₂²×C₇×C₈	D₄	1	2	yes	—
31	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
32	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
33	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
34	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
35	C₂²×C₇×C₈	Q₈	1	2	yes	—
36	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
37	C₂×C₄×Dic₇	—	2	1	no	—
38	C₂×C₄×Dic₇	—	2	1	no	—
39	C₂×C₄×Dic₇	—	2	1	no	—
40	C₂×C₄×Dic₇	—	2	1	no	—
41	C₂×C₄×Dic₇	C₈	1	2	yes	—
42	C₂×C₄×Dic₇	C₂×C₄	1	2	yes	—
43	C₂²×C₇⋊C₈	—	2	1	no	—
44	C₂²×C₇⋊C₈	—	2	1	no	—
45	C₂²×C₇⋊C₈	—	2	2	no	—
46	C₂²×C₇⋊C₈	C₈	1	4	yes	—
47	C₂²×C₇⋊C₈	C₂×C₄	1	2	yes	—
48	C₇×M₃₂	—	2	4	no	—
49	C₇×M₃₂	—	2	2	no	—
50	C₇×M₃₂	C₂×C₄	1	4	yes	complemented
51	C₂×C₇×C₁₆	—	2	4	no	—
52	C₂×C₇×C₁₆	—	2	2	no	—
53	#6389	—	2	2	no	—
54	#6389	—	2	2	no	—
55	#6389	—	2	2	no	—
56	#6389	—	2	2	no	—
57	#6386	—	2	4	no	—
58	#6386	—	2	2	no	—
59	#6386	C₂×C₄	1	4	yes	complemented
60	C₂×C₇⋊C₁₆	—	2	4	no	—
61	C₂×C₇⋊C₁₆	—	2	2	no	—
62	C₂²×M₃₂	—	7	1	no	—
63	C₂²×M₃₂	D₇	1	1	yes	characteristic
64	C₂²×#97	—	7	1	no	—
65	#1377	—	7	4	no	—
66	C₂⁴×C₇	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₇	C₄²	1	2	yes	—
68	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
73	C₂²×C₄×C₇	M₁₆	1	2	yes	—
74	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
75	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
76	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×Dic₇	—	2	2	no	—
78	C₂²×Dic₇	—	2	2	no	—
79	C₂²×Dic₇	—	4	4	no	—
80	C₂²×Dic₇	C₂×C₈	1	4	yes	—
81	C₂×C₇×C₈	C₄²	1	2	yes	—
82	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
85	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
86	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
87	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
88	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
89	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
90	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
91	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
92	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
93	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
96	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
98	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
99	C₄×Dic₇	—	2	2	no	—
100	C₄×Dic₇	—	2	2	no	—
101	C₄×Dic₇	—	2	2	no	—
102	C₄×Dic₇	—	2	2	no	—
103	C₂×C₇⋊C₈	—	2	4	no	—
104	C₂×C₇⋊C₈	—	2	4	no	—
105	C₂×C₇⋊C₈	—	2	2	no	—
106	C₂×C₇⋊C₈	—	2	2	no	—
107	C₂×C₇⋊C₈	C₂×C₈	1	4	yes	—
108	C₇×C₁₆	—	2	4	no	—
109	C₇⋊C₁₆	—	2	4	no	—
110	C₂³×C₈	—	7	1	no	—
111	C₂³×C₈	C₂×D₇	1	1	yes	characteristic
112	C₂²×C₄²	—	7	1	no	—
113	C₂×M₃₂	—	2	2	no	—
114	C₂×M₃₂	—	2	1	no	—
115	C₂×M₃₂	—	7	4	no	—
116	C₂×M₃₂	—	7	2	no	—
117	C₂×M₃₂	—	14	1	no	—
118	C₂×M₃₂	—	14	2	no	—
119	C₂×M₃₂	Dic₇	1	4	yes	complemented
120	C₂×M₃₂	C₂×D₇	1	2	yes	—
121	C₂²×C₁₆	—	2	1	no	—
122	C₂²×C₁₆	—	14	1	no	—
123	C₂×#97	—	7	2	no	—
124	C₂×#97	—	7	2	no	—
125	C₂×#97	—	14	1	no	—
126	C₂×#97	—	14	1	no	—
127	C₂×#97	—	14	1	no	—
128	C₂×#97	—	14	1	no	—
129	C₂³×C₇	—	2	4	no	—
130	C₂³×C₇	C₄×C₈	1	2	yes	—
131	C₂³×C₇	C₂²⋊C₈	1	1	yes	characteristic
132	C₂³×C₇	C₂²⋊C₈	1	1	yes	characteristic
133	C₂³×C₇	C₄⋊C₈	1	2	yes	—
134	C₂³×C₇	C₂²×C₈	1	1	yes	characteristic
135	C₂×C₄×C₇	#95	1	2	yes	—
136	C₂×C₄×C₇	#95	1	2	yes	—
137	C₂×C₄×C₇	C₄×C₈	1	2	yes	—
138	C₂×C₄×C₇	#97	1	2	yes	—
139	C₂×C₄×C₇	#97	1	2	yes	—
140	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
141	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
142	C₂×C₄×C₇	C₂²⋊C₈	1	2	yes	—
143	C₂×C₄×C₇	C₂²⋊C₈	1	2	yes	—
144	C₂×C₄×C₇	C₄⋊C₈	1	2	yes	—
145	C₂×C₄×C₇	C₄⋊C₈	1	2	yes	—
146	C₂×C₄×C₇	C₄⋊C₈	1	2	yes	—
147	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
148	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
151	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
152	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
153	C₂×Dic₇	—	2	4	no	—
154	C₂×Dic₇	—	4	4	no	—
155	C₂×Dic₇	—	4	4	no	—
156	C₂×Dic₇	—	4	4	no	—
157	C₇×C₈	#95	1	2	yes	—
158	C₇×C₈	#95	1	2	yes	—
159	C₇×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
160	C₇×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
161	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
162	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
163	C₇⋊C₈	—	2	4	no	—
164	C₂³×C₄	—	7	2	no	—
165	C₂³×C₄	C₄×D₇	1	1	yes	characteristic
166	C₂²×C₈	—	7	4	no	—
167	C₂²×C₈	—	7	2	no	—
168	C₂²×C₈	—	14	2	no	—
169	C₂²×C₈	—	14	1	no	—
170	C₂²×C₈	—	14	1	no	—
171	C₂²×C₈	Dic₁₄	1	2	yes	—
172	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
173	C₂²×C₈	C₄×D₇	1	2	yes	—
174	C₂²×C₈	D₂₈	1	1	yes	characteristic
175	C₂²×C₈	D₂₈	1	1	yes	characteristic
176	C₂²×C₈	C₂×Dic₇	1	2	yes	—
177	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
178	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
179	C₂²×C₈	C₂²⋊D₇	1	2	yes	—
180	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
181	C₂×C₄²	—	7	2	no	—
182	C₂×C₄²	—	7	2	no	—
183	C₂×C₄²	—	14	1	no	—
184	C₂×C₄²	—	14	1	no	—
185	C₂×C₄²	—	14	1	no	—
186	C₂×C₄²	—	14	1	no	—
187	M₃₂	—	2	4	no	—
188	M₃₂	—	2	2	no	—
189	M₃₂	—	7	4	no	—
190	M₃₂	—	14	4	no	—
191	M₃₂	—	14	2	no	—
192	M₃₂	C₂×Dic₇	1	4	yes	complemented
193	C₂×C₁₆	—	2	4	no	—
194	C₂×C₁₆	—	2	2	no	—
195	C₂×C₁₆	—	14	4	no	—
196	C₂×C₁₆	—	14	2	no	—
197	#97	—	14	2	no	—
198	#97	—	14	2	no	—
199	#97	—	14	2	no	—
200	#97	—	14	2	no	—
201	C₂²×C₇	—	2	2	no	—
202	C₂²×C₇	—	2	2	no	—
203	C₂²×C₇	—	2	4	no	—
204	C₂²×C₇	—	2	4	no	—
205	C₂²×C₇	#336	1	2	yes	—
206	C₂²×C₇	#336	1	2	yes	—
207	C₂²×C₇	#349	1	4	yes	—
208	C₂²×C₇	C₂×C₄×C₈	1	1	yes	characteristic
209	C₂²×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
210	C₂²×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
211	C₄×C₇	#336	1	2	yes	—
212	C₄×C₇	#336	1	2	yes	—
213	C₄×C₇	C₂×#95	1	1	yes	characteristic
214	C₄×C₇	C₂×#97	1	1	yes	characteristic
215	C₄×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
216	C₄×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
217	Dic₇	—	4	4	no	—
218	C₂⁴	C₈×D₇	1	1	yes	characteristic
219	C₂²×C₄	—	7	4	no	—
220	C₂²×C₄	—	14	2	no	—
221	C₂²×C₄	—	14	2	no	—
222	C₂²×C₄	—	28	4	no	—
223	C₂²×C₄	C₈×D₇	1	2	yes	—
224	C₂²×C₄	C₈×D₇	1	1	yes	characteristic
225	C₂²×C₄	#1131	1	1	yes	characteristic
226	C₂²×C₄	#1131	1	1	yes	characteristic
227	C₂²×C₄	#1131	1	2	yes	—
228	C₂²×C₄	C₄×Dic₇	1	2	yes	—
229	C₂²×C₄	Dic₇⋊C₄	1	2	yes	—
230	C₂²×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
231	C₂²×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
233	C₂×C₈	—	7	4	no	—
234	C₂×C₈	—	14	2	no	—
235	C₂×C₈	—	14	2	no	—
236	C₂×C₈	—	14	4	no	—
237	C₂×C₈	—	14	4	no	—
238	C₂×C₈	C₄×Dic₇	1	2	yes	—
239	C₂×C₈	Dic₇⋊C₄	1	2	yes	—
240	C₂×C₈	Dic₇⋊C₄	1	2	yes	—
241	C₂×C₈	Dic₇⋊C₄	1	2	yes	—
242	C₂×C₈	C₄⋊Dic₇	1	2	yes	—
243	C₂×C₈	C₄⋊Dic₇	1	2	yes	—
244	C₂×C₈	(C₂×D₇)⋊C₄	1	1	yes	characteristic
245	C₂×C₈	(C₂×D₇)⋊C₄	1	1	yes	characteristic
246	C₂×C₈	(C₂×D₇)⋊C₄	1	2	yes	—
247	C₂×C₈	(C₂×D₇)⋊C₄	1	2	yes	—
248	C₂×C₈	C₂²⋊Dic₇	1	2	yes	—
249	C₂×C₈	C₂²⋊Dic₇	1	2	yes	—
250	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
251	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
252	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
253	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
254	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
255	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
256	C₄²	—	14	2	no	—
257	C₄²	—	14	2	no	—
258	C₄²	—	14	2	no	—
259	C₄²	—	14	2	no	—
260	C₁₆	—	2	4	no	—
261	C₁₆	—	14	4	no	—
262	C₂×C₇	—	2	2	no	—
263	C₂×C₇	—	2	1	no	—
264	C₂×C₇	—	2	1	no	—
265	C₂×C₇	#1377	1	4	yes	complemented
266	C₂×C₇	C₂×#336	1	1	yes	characteristic
267	C₂×C₇	C₂×#349	1	2	yes	—
268	C₂³	—	2	4	no	—
269	C₂³	C₈×Dic₇	1	2	yes	—
270	C₂³	Dic₇⋊C₈	1	2	yes	—
271	C₂³	(C₂×D₇)⋊C₈	1	1	yes	characteristic
272	C₂³	(C₂×D₇)⋊C₈	1	1	yes	characteristic
273	C₂³	C₂×C₈×D₇	1	1	yes	characteristic
274	C₂×C₄	—	14	4	no	—
275	C₂×C₄	—	28	4	no	—
276	C₂×C₄	—	28	4	no	—
277	C₂×C₄	—	28	4	no	—
278	C₂×C₄	C₈×Dic₇	1	2	yes	—
279	C₂×C₄	Dic₇⋊C₈	1	2	yes	—
280	C₂×C₄	Dic₇⋊C₈	1	2	yes	—
281	C₂×C₄	Dic₇⋊C₈	1	2	yes	—
282	C₂×C₄	#6389	1	2	yes	—
283	C₂×C₄	#6389	1	2	yes	—
284	C₂×C₄	(C₂×D₇)⋊C₈	1	1	yes	characteristic
285	C₂×C₄	(C₂×D₇)⋊C₈	1	1	yes	characteristic
286	C₂×C₄	(C₂×D₇)⋊C₈	1	2	yes	—
287	C₂×C₄	(C₂×D₇)⋊C₈	1	2	yes	—
288	C₂×C₄	C₇⋊#95	1	2	yes	—
289	C₂×C₄	C₇⋊#95	1	2	yes	—
290	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
291	C₂×C₄	C₂×#1131	1	1	yes	characteristic
292	C₂×C₄	C₂×#1131	1	1	yes	characteristic
293	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
294	C₂×C₄	C₂×Dic₇⋊C₄	1	1	yes	characteristic
295	C₂×C₄	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
296	C₈	—	14	4	no	—
297	C₈	C₇⋊#95	1	2	yes	—
298	C₈	C₇⋊#95	1	2	yes	—
299	C₈	C₂×Dic₇⋊C₄	1	1	yes	characteristic
300	C₈	C₂×C₄⋊Dic₇	1	1	yes	characteristic
301	C₈	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
302	C₈	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
303	C₇	C₂×#1377	1	1	yes	characteristic, complemented, Sylow
304	C₂²	—	2	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	4	no	—
307	C₂²	—	2	4	no	—
308	C₂²	#89309	1	4	yes	—
309	C₂²	#89345	1	2	yes	—
310	C₂²	#89345	1	2	yes	—
311	C₂²	C₂×C₈×Dic₇	1	1	yes	characteristic
312	C₂²	C₂×Dic₇⋊C₈	1	1	yes	characteristic
313	C₂²	C₂×(C₂×D₇)⋊C₈	1	1	yes	characteristic
314	C₄	—	28	4	no	—
315	C₄	#89345	1	2	yes	—
316	C₄	#89345	1	2	yes	—
317	C₄	C₂×Dic₇⋊C₈	1	1	yes	characteristic
318	C₄	C₂×#6389	1	1	yes	characteristic
319	C₄	C₂×(C₂×D₇)⋊C₈	1	1	yes	characteristic
320	C₄	C₂×C₇⋊#95	1	1	yes	characteristic
321	C₂	—	2	2	no	—
322	C₂	—	2	1	no	—
323	C₂	—	2	1	no	—
324	C₂	#11725618	1	4	yes	direct product factor
325	C₂	C₂×#89309	1	2	yes	—
326	C₂	C₂×#89345	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	(C₂×D₇)⋊C₄
commutator	C₂²×C₇	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₂²×C₇×M₃₂	C₂
socle	C₂³×C₇	C₂²⋊C₈
prime radical	C₂⁴×C₇	C₂×C₈
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	C₂²×M₃₂	D₇
hypocenter	C₇	C₂×#1377
LCS	G ▷ C₂²×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
4. #11725618	C₂	direct (N × H)
10. C₂×C₇×M₃₂	C₄	semidirect
21. C₂×#6386	C₄	semidirect
50. C₇×M₃₂	C₂×C₄	semidirect
59. #6386	C₂×C₄	semidirect
119. C₂×M₃₂	Dic₇	semidirect
192. M₃₂	C₂×Dic₇	semidirect
265. C₂×C₇	#1377	semidirect
303. C₇	C₂×#1377	semidirect
324. C₂	#11725618	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₇×M₃₂	2
C₂²×#6389	2
C₂²×#6386	2
#11725618	2
C₂×#1377	7

Subgroups — C2×#11725618 #13268654

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11725618 #13268654