Subgroups — C₂×#11725616 #13268648

Order 1792.

670 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	C₂²×C₇×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×Dic₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×#6386	C₂	1	1	yes	maximal, characteristic
4	#11725616	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×C₈×Dic₇	—	2	2	no	—
14	C₂×C₈×Dic₇	—	2	2	no	—
15	C₂×C₈×Dic₇	C₄	1	2	yes	—
16	C₂×C₈×Dic₇	C₂²	1	2	yes	—
17	C₂×#6386	—	2	1	no	—
18	C₂×#6386	—	2	2	no	—
19	C₂×#6386	C₄	1	4	yes	complemented
20	C₂×#6386	C₂²	1	2	yes	—
21	C₂²×C₇⋊C₁₆	—	2	1	no	—
22	C₂×#1376	—	7	1	no	maximal, Sylow
23	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
24	C₂³×Dic₇	—	2	1	no	—
25	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
27	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
28	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
29	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
30	C₂²×C₇×C₈	D₄	1	2	yes	—
31	C₂²×C₇×C₈	D₄	1	2	yes	—
32	C₂²×C₇×C₈	Q₈	1	1	yes	characteristic
33	C₂²×C₇×C₈	Q₈	1	1	yes	characteristic
34	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
35	C₂×C₄×Dic₇	—	2	2	no	—
36	C₂×C₄×Dic₇	—	2	1	no	—
37	C₂×C₄×Dic₇	C₈	1	4	yes	—
38	C₂×C₄×Dic₇	C₂×C₄	1	2	yes	—
39	C₂²×C₇⋊C₈	—	2	2	no	—
40	C₂²×C₇⋊C₈	—	2	2	no	—
41	C₂²×C₇⋊C₈	C₈	1	2	yes	—
42	C₂²×C₇⋊C₈	C₈	1	2	yes	—
43	C₂²×C₇⋊C₈	C₂×C₄	1	2	yes	—
44	C₇×M₃₂	—	2	4	no	—
45	C₇×M₃₂	—	2	2	no	—
46	C₇×M₃₂	C₂×C₄	1	4	yes	complemented
47	C₂×C₇×C₁₆	—	2	4	no	—
48	C₂×C₇×C₁₆	—	2	2	no	—
49	C₈×Dic₇	—	2	4	no	—
50	C₈×Dic₇	—	2	2	no	—
51	C₈×Dic₇	—	2	2	no	—
52	#6386	—	2	4	no	—
53	#6386	—	2	2	no	—
54	#6386	C₂×C₄	1	4	yes	complemented
55	C₂×C₇⋊C₁₆	—	2	4	no	—
56	C₂×C₇⋊C₁₆	—	2	2	no	—
57	C₂²×M₃₂	—	7	1	no	—
58	C₂²×M₃₂	D₇	1	1	yes	characteristic
59	C₂²×C₄×C₈	—	7	1	no	—
60	#1376	—	7	4	no	—
61	C₂⁴×C₇	M₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₇	C₄²	1	2	yes	—
63	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
68	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
69	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
70	C₂²×C₄×C₇	M₁₆	1	2	yes	—
71	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×Dic₇	—	2	8	no	—
73	C₂²×Dic₇	—	2	4	no	—
74	C₂²×Dic₇	—	2	2	no	—
75	C₂×C₇×C₈	C₄²	1	2	yes	—
76	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
77	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
78	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
79	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
80	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
81	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
84	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
85	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
86	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
87	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
90	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
91	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
92	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
93	C₄×Dic₇	—	2	4	no	—
94	C₄×Dic₇	—	2	2	no	—
95	C₄×Dic₇	C₂×C₈	1	4	yes	—
96	C₂×C₇⋊C₈	—	2	4	no	—
97	C₂×C₇⋊C₈	—	2	4	no	—
98	C₂×C₇⋊C₈	—	2	2	no	—
99	C₂×C₇⋊C₈	—	2	2	no	—
100	C₂×C₇⋊C₈	C₂×C₈	1	4	yes	—
101	C₇×C₁₆	—	2	4	no	—
102	C₇⋊C₁₆	—	2	4	no	—
103	C₂³×C₈	—	7	1	no	—
104	C₂³×C₈	C₂×D₇	1	1	yes	characteristic
105	C₂²×C₄²	—	7	1	no	—
106	C₂×M₃₂	—	2	2	no	—
107	C₂×M₃₂	—	2	1	no	—
108	C₂×M₃₂	—	7	4	no	—
109	C₂×M₃₂	—	7	2	no	—
110	C₂×M₃₂	—	14	2	no	—
111	C₂×M₃₂	—	14	1	no	—
112	C₂×M₃₂	Dic₇	1	4	yes	complemented
113	C₂×M₃₂	C₂×D₇	1	2	yes	—
114	C₂²×C₁₆	—	2	1	no	—
115	C₂²×C₁₆	—	14	1	no	—
116	C₂×C₄×C₈	—	7	2	no	—
117	C₂×C₄×C₈	—	7	2	no	—
118	C₂×C₄×C₈	—	14	2	no	—
119	C₂×C₄×C₈	—	14	2	no	—
120	C₂³×C₇	—	2	4	no	—
121	C₂³×C₇	#97	1	2	yes	—
122	C₂³×C₇	C₂²⋊C₈	1	2	yes	—
123	C₂³×C₇	C₄⋊C₈	1	1	yes	characteristic
124	C₂³×C₇	C₄⋊C₈	1	1	yes	characteristic
125	C₂³×C₇	C₂×M₁₆	1	1	yes	characteristic
126	C₂×C₄×C₇	#95	1	2	yes	—
127	C₂×C₄×C₇	#95	1	2	yes	—
128	C₂×C₄×C₇	C₄×C₈	1	2	yes	—
129	C₂×C₄×C₇	C₄×C₈	1	2	yes	—
130	C₂×C₄×C₇	#97	1	2	yes	—
131	C₂×C₄×C₇	C₂²⋊C₈	1	2	yes	—
132	C₂×C₄×C₇	C₂²⋊C₈	1	2	yes	—
133	C₂×C₄×C₇	C₂²⋊C₈	1	2	yes	—
134	C₂×C₄×C₇	C₄⋊C₈	1	1	yes	characteristic
135	C₂×C₄×C₇	C₄⋊C₈	1	1	yes	characteristic
136	C₂×C₄×C₇	C₄⋊C₈	1	2	yes	—
137	C₂×C₄×C₇	C₄⋊C₈	1	2	yes	—
138	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
139	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
142	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
143	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
144	C₂×Dic₇	—	2	16	no	—
145	C₂×Dic₇	—	2	8	no	—
146	C₂×Dic₇	—	2	4	no	—
147	C₇×C₈	#95	1	2	yes	—
148	C₇×C₈	#95	1	2	yes	—
149	C₇×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
150	C₇×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
151	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₇⋊C₈	—	2	4	no	—
154	C₂³×C₄	—	14	1	no	—
155	C₂³×C₄	C₄×D₇	1	1	yes	characteristic
156	C₂²×C₈	—	7	2	no	—
157	C₂²×C₈	—	7	2	no	—
158	C₂²×C₈	—	7	2	no	—
159	C₂²×C₈	—	14	2	no	—
160	C₂²×C₈	—	14	2	no	—
161	C₂²×C₈	Dic₁₄	1	1	yes	characteristic
162	C₂²×C₈	Dic₁₄	1	1	yes	characteristic
163	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
164	C₂²×C₈	C₄×D₇	1	2	yes	—
165	C₂²×C₈	D₂₈	1	2	yes	—
166	C₂²×C₈	C₂×Dic₇	1	2	yes	—
167	C₂²×C₈	C₂²⋊D₇	1	2	yes	—
168	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
169	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
170	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
171	C₂×C₄²	—	7	4	no	—
172	C₂×C₄²	—	7	2	no	—
173	C₂×C₄²	—	14	1	no	—
174	C₂×C₄²	—	14	2	no	—
175	M₃₂	—	2	4	no	—
176	M₃₂	—	2	2	no	—
177	M₃₂	—	7	4	no	—
178	M₃₂	—	14	4	no	—
179	M₃₂	—	14	2	no	—
180	M₃₂	C₂×Dic₇	1	4	yes	complemented
181	C₂×C₁₆	—	2	4	no	—
182	C₂×C₁₆	—	2	2	no	—
183	C₂×C₁₆	—	14	4	no	—
184	C₂×C₁₆	—	14	2	no	—
185	C₄×C₈	—	14	4	no	—
186	C₄×C₈	—	14	2	no	—
187	C₄×C₈	—	14	2	no	—
188	C₂²×C₇	—	2	4	no	—
189	C₂²×C₇	—	2	4	no	—
190	C₂²×C₇	—	2	2	no	—
191	C₂²×C₇	—	2	2	no	—
192	C₂²×C₇	#336	1	2	yes	—
193	C₂²×C₇	#336	1	2	yes	—
194	C₂²×C₇	#364	1	4	yes	—
195	C₂²×C₇	C₂×#97	1	1	yes	characteristic
196	C₂²×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
197	C₂²×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
198	C₄×C₇	#336	1	2	yes	—
199	C₄×C₇	#336	1	2	yes	—
200	C₄×C₇	C₂×#95	1	1	yes	characteristic
201	C₄×C₇	C₂×C₄×C₈	1	1	yes	characteristic
202	C₄×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
203	C₄×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
204	Dic₇	—	2	8	no	—
205	C₂⁴	#1131	1	1	yes	characteristic
206	C₂²×C₄	—	14	8	no	—
207	C₂²×C₄	—	14	2	no	—
208	C₂²×C₄	—	14	4	no	—
209	C₂²×C₄	C₈×D₇	1	2	yes	—
210	C₂²×C₄	C₈×D₇	1	1	yes	characteristic
211	C₂²×C₄	C₈×D₇	1	1	yes	characteristic
212	C₂²×C₄	#1131	1	2	yes	—
213	C₂²×C₄	#1131	1	1	yes	characteristic
214	C₂²×C₄	C₄×Dic₇	1	2	yes	—
215	C₂²×C₄	Dic₇⋊C₄	1	1	yes	characteristic
216	C₂²×C₄	Dic₇⋊C₄	1	1	yes	characteristic
217	C₂²×C₄	(C₂×D₇)⋊C₄	1	2	yes	—
218	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
219	C₂×C₈	—	7	4	no	—
220	C₂×C₈	—	14	2	no	—
221	C₂×C₈	—	14	2	no	—
222	C₂×C₈	—	14	4	no	—
223	C₂×C₈	—	14	4	no	—
224	C₂×C₈	C₄×Dic₇	1	2	yes	—
225	C₂×C₈	Dic₇⋊C₄	1	1	yes	characteristic
226	C₂×C₈	Dic₇⋊C₄	1	1	yes	characteristic
227	C₂×C₈	Dic₇⋊C₄	1	2	yes	—
228	C₂×C₈	Dic₇⋊C₄	1	2	yes	—
229	C₂×C₈	C₄⋊Dic₇	1	2	yes	—
230	C₂×C₈	C₄⋊Dic₇	1	2	yes	—
231	C₂×C₈	(C₂×D₇)⋊C₄	1	2	yes	—
232	C₂×C₈	(C₂×D₇)⋊C₄	1	2	yes	—
233	C₂×C₈	(C₂×D₇)⋊C₄	1	2	yes	—
234	C₂×C₈	C₂²⋊Dic₇	1	2	yes	—
235	C₂×C₈	C₂²⋊Dic₇	1	2	yes	—
236	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
237	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
238	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
239	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
240	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
241	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
242	C₄²	—	7	4	no	—
243	C₄²	—	14	2	no	—
244	C₄²	—	14	4	no	—
245	C₁₆	—	2	4	no	—
246	C₁₆	—	14	4	no	—
247	C₂×C₇	—	2	1	no	—
248	C₂×C₇	—	2	2	no	—
249	C₂×C₇	—	2	1	no	—
250	C₂×C₇	#1376	1	4	yes	complemented
251	C₂×C₇	C₂×#336	1	1	yes	characteristic
252	C₂×C₇	C₂×#364	1	2	yes	—
253	C₂³	—	2	4	no	—
254	C₂³	Dic₇⋊C₈	1	1	yes	characteristic
255	C₂³	Dic₇⋊C₈	1	1	yes	characteristic
256	C₂³	#6389	1	2	yes	—
257	C₂³	(C₂×D₇)⋊C₈	1	2	yes	—
258	C₂³	C₂×#1131	1	1	yes	characteristic
259	C₂×C₄	—	14	8	no	—
260	C₂×C₄	—	14	16	no	—
261	C₂×C₄	—	14	4	no	—
262	C₂×C₄	C₈×Dic₇	1	2	yes	—
263	C₂×C₄	C₈×Dic₇	1	2	yes	—
264	C₂×C₄	Dic₇⋊C₈	1	1	yes	characteristic
265	C₂×C₄	Dic₇⋊C₈	1	1	yes	characteristic
266	C₂×C₄	Dic₇⋊C₈	1	2	yes	—
267	C₂×C₄	Dic₇⋊C₈	1	2	yes	—
268	C₂×C₄	#6389	1	2	yes	—
269	C₂×C₄	(C₂×D₇)⋊C₈	1	2	yes	—
270	C₂×C₄	(C₂×D₇)⋊C₈	1	2	yes	—
271	C₂×C₄	(C₂×D₇)⋊C₈	1	2	yes	—
272	C₂×C₄	C₇⋊#95	1	2	yes	—
273	C₂×C₄	C₇⋊#95	1	2	yes	—
274	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
275	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
276	C₂×C₄	C₂×#1131	1	1	yes	characteristic
277	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
278	C₂×C₄	C₂×Dic₇⋊C₄	1	1	yes	characteristic
279	C₂×C₄	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
280	C₈	—	14	4	no	—
281	C₈	C₇⋊#95	1	2	yes	—
282	C₈	C₇⋊#95	1	2	yes	—
283	C₈	C₂×Dic₇⋊C₄	1	1	yes	characteristic
284	C₈	C₂×C₄⋊Dic₇	1	1	yes	characteristic
285	C₈	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
286	C₈	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
287	C₇	C₂×#1376	1	1	yes	characteristic, complemented, Sylow
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	4	no	—
291	C₂²	—	2	4	no	—
292	C₂²	#89306	1	4	yes	—
293	C₂²	#89345	1	2	yes	—
294	C₂²	#89345	1	2	yes	—
295	C₂²	C₂×Dic₇⋊C₈	1	1	yes	characteristic
296	C₂²	C₂×#6389	1	1	yes	characteristic
297	C₂²	C₂×(C₂×D₇)⋊C₈	1	1	yes	characteristic
298	C₄	—	14	8	no	—
299	C₄	#89345	1	2	yes	—
300	C₄	#89345	1	2	yes	—
301	C₄	C₂×C₈×Dic₇	1	1	yes	characteristic
302	C₄	C₂×Dic₇⋊C₈	1	1	yes	characteristic
303	C₄	C₂×(C₂×D₇)⋊C₈	1	1	yes	characteristic
304	C₄	C₂×C₇⋊#95	1	1	yes	characteristic
305	C₂	—	2	1	no	—
306	C₂	—	2	2	no	—
307	C₂	—	2	1	no	—
308	C₂	#11725616	1	4	yes	direct product factor
309	C₂	C₂×#89306	1	2	yes	—
310	C₂	C₂×#89345	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²⋊D₇
commutator	C₄×C₇	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₂²×C₇×M₃₂	C₂
socle	C₂³×C₇	C₄⋊C₈
prime radical	C₂⁴×C₇	M₁₆
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	C₂²×M₃₂	D₇
hypocenter	C₇	C₂×#1376
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
4. #11725616	C₂	direct (N × H)
10. C₂×C₇×M₃₂	C₄	semidirect
19. C₂×#6386	C₄	semidirect
46. C₇×M₃₂	C₂×C₄	semidirect
54. #6386	C₂×C₄	semidirect
112. C₂×M₃₂	Dic₇	semidirect
180. M₃₂	C₂×Dic₇	semidirect
250. C₂×C₇	#1376	semidirect
287. C₇	C₂×#1376	semidirect
308. C₂	#11725616	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×C₈×Dic₇	2
C₂²×#6386	2
#11725616	2
C₂×#1376	7

Subgroups — C2×#11725616 #13268648

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11725616 #13268648