Subgroups — C₂×#11725626 #13268708

Order 1792.

554 conjugacy classes of subgroups.

The subgroup lattice has 270 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×#6393	C₂	1	2	yes	maximal
3	#11725626	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
5	C₂²×#1136	—	2	2	no	—
6	C₂×C₇×M₃₂	—	2	2	no	—
7	C₂×C₇×M₃₂	—	2	2	no	—
8	C₂×C₇×M₃₂	C₄	1	2	yes	—
9	C₂×C₇×M₃₂	C₂²	1	2	yes	—
10	C₂²×C₇×C₁₆	C₄	1	2	yes	—
11	C₂×#6393	—	2	2	no	—
12	C₂×#6393	—	2	2	no	—
13	C₂×#6393	—	2	2	no	—
14	C₂×#6393	C₄	1	4	yes	—
15	C₂×#6393	C₄	1	4	yes	—
16	C₂×#6393	C₂²	1	4	yes	—
17	C₂×#1385	—	7	1	no	maximal, Sylow
18	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
19	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
22	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
23	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
24	C₂²×C₇×C₈	D₄	1	2	yes	—
25	C₂²×C₇×C₈	D₄	1	2	yes	—
26	C₂²×C₇×C₈	Q₈	1	1	yes	characteristic
27	C₂²×C₇×C₈	Q₈	1	1	yes	characteristic
28	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
29	C₂×#1136	—	2	4	no	—
30	C₂×#1136	—	2	4	no	—
31	C₂×#1136	—	2	4	no	—
32	C₂×#1136	—	4	2	no	—
33	C₂×#1136	—	4	2	no	—
34	C₂×#1136	—	4	2	no	—
35	C₂²×C₇⋊C₈	—	4	2	no	—
36	C₇×M₃₂	—	2	4	no	—
37	C₇×M₃₂	—	2	4	no	—
38	C₂×C₇×C₁₆	—	2	2	no	—
39	C₂×C₇×C₁₆	—	2	2	no	—
40	C₂×C₇×C₁₆	C₂×C₄	1	4	yes	—
41	#6393	—	2	4	no	—
42	#6393	—	2	4	no	—
43	#6393	—	2	4	no	—
44	#6393	C₂×C₄	1	8	yes	—
45	C₂²×M₃₂	D₇	1	1	yes	characteristic
46	C₂²×#108	—	7	2	no	—
47	#1385	—	7	4	no	—
48	C₂⁴×C₇	Q₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₇	SD₁₆	1	2	yes	—
56	C₂²×C₄×C₇	SD₁₆	1	2	yes	—
57	C₂²×C₄×C₇	Q₁₆	1	1	yes	characteristic
58	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
59	C₂×C₇×C₈	C₄²	1	1	yes	characteristic
60	C₂×C₇×C₈	C₄²	1	1	yes	characteristic
61	C₂×C₇×C₈	C₄²	1	2	yes	—
62	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
63	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
64	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
67	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
70	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
71	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
72	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
73	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
75	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
76	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
77	#1136	—	2	8	no	—
78	#1136	—	4	4	no	—
79	#1136	—	4	4	no	—
80	#1136	—	4	4	no	—
81	C₂×C₇⋊C₈	—	4	4	no	—
82	C₂×C₇⋊C₈	—	4	4	no	—
83	C₂×C₇⋊C₈	—	4	4	no	—
84	C₇×C₁₆	—	2	4	no	—
85	C₂²×M₁₆	—	14	2	no	—
86	C₂³×C₈	C₂×D₇	1	1	yes	characteristic
87	C₂×M₃₂	—	2	2	no	—
88	C₂×M₃₂	—	2	2	no	—
89	C₂×M₃₂	Dic₇	1	2	yes	—
90	C₂×M₃₂	C₂×D₇	1	2	yes	—
91	C₂²×C₁₆	Dic₇	1	2	yes	—
92	C₂×#108	—	7	4	no	—
93	C₂×#108	—	7	4	no	—
94	C₂×#108	—	7	4	no	—
95	C₂×#108	—	14	2	no	—
96	C₂×#108	—	14	2	no	—
97	C₂×#108	—	14	2	no	—
98	C₂³×C₇	—	4	2	no	—
99	C₂³×C₇	#103	1	2	yes	—
100	C₂³×C₇	#103	1	2	yes	—
101	C₂³×C₇	#107	1	1	yes	characteristic
102	C₂³×C₇	#107	1	1	yes	characteristic
103	C₂³×C₇	C₂×Q₁₆	1	1	yes	characteristic
104	C₂×C₄×C₇	#95	1	1	yes	characteristic
105	C₂×C₄×C₇	#95	1	2	yes	—
106	C₂×C₄×C₇	#95	1	1	yes	characteristic
107	C₂×C₄×C₇	#102	1	2	yes	—
108	C₂×C₄×C₇	#102	1	2	yes	—
109	C₂×C₄×C₇	#102	1	2	yes	—
110	C₂×C₄×C₇	#102	1	2	yes	—
111	C₂×C₄×C₇	#103	1	2	yes	—
112	C₂×C₄×C₇	#103	1	2	yes	—
113	C₂×C₄×C₇	#106	1	2	yes	—
114	C₂×C₄×C₇	#106	1	1	yes	characteristic
115	C₂×C₄×C₇	#106	1	1	yes	characteristic
116	C₂×C₄×C₇	#107	1	1	yes	characteristic
117	C₂×C₄×C₇	#107	1	1	yes	characteristic
118	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
119	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
122	C₇×C₈	#95	1	1	yes	characteristic
123	C₇×C₈	#95	1	1	yes	characteristic
124	C₇×C₈	#95	1	2	yes	—
125	C₇×C₈	C₂×C₄²	1	1	yes	characteristic
126	C₇×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₇×C₈	C₂×C₄⋊C₄	1	2	yes	—
128	C₇⋊C₈	—	4	8	no	—
129	C₂³×C₄	D₂₈	1	1	yes	characteristic
130	C₂×M₁₆	—	14	4	no	—
131	C₂×M₁₆	—	14	4	no	—
132	C₂×M₁₆	—	14	4	no	—
133	C₂×M₁₆	—	28	2	no	—
134	C₂×M₁₆	—	28	2	no	—
135	C₂×M₁₆	—	28	2	no	—
136	C₂²×C₈	—	28	2	no	—
137	C₂²×C₈	Dic₁₄	1	1	yes	characteristic
138	C₂²×C₈	Dic₁₄	1	1	yes	characteristic
139	C₂²×C₈	C₄×D₇	1	2	yes	—
140	C₂²×C₈	C₄×D₇	1	2	yes	—
141	C₂²×C₈	D₂₈	1	1	yes	characteristic
142	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
143	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
144	C₂²×C₈	C₂²⋊D₇	1	2	yes	—
145	C₂²×C₈	C₂²⋊D₇	1	2	yes	—
146	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
147	M₃₂	—	2	4	no	—
148	M₃₂	—	2	4	no	—
149	C₂×C₁₆	—	2	2	no	—
150	C₂×C₁₆	—	2	2	no	—
151	C₂×C₁₆	C₂×Dic₇	1	4	yes	—
152	#108	—	7	8	no	—
153	#108	—	14	4	no	—
154	#108	—	14	4	no	—
155	#108	—	14	4	no	—
156	C₂²×C₇	—	2	4	no	—
157	C₂²×C₇	—	2	2	no	—
158	C₂²×C₇	—	4	2	no	—
159	C₂²×C₇	—	4	1	no	—
160	C₂²×C₇	—	4	1	no	—
161	C₂²×C₇	#340	1	2	yes	—
162	C₂²×C₇	#340	1	1	yes	characteristic
163	C₂²×C₇	#340	1	1	yes	characteristic
164	C₂²×C₇	C₂×#103	1	2	yes	—
165	C₂²×C₇	C₂×#107	1	1	yes	characteristic
166	C₄×C₇	#340	1	2	yes	—
167	C₄×C₇	#340	1	1	yes	characteristic
168	C₄×C₇	#340	1	1	yes	characteristic
169	C₄×C₇	C₂×#95	1	1	yes	characteristic
170	C₄×C₇	C₂×#102	1	2	yes	—
171	C₄×C₇	C₂×#106	1	1	yes	characteristic
172	C₂⁴	Dic₂₈	1	1	yes	characteristic
173	C₂²×C₄	#1132	1	2	yes	—
174	C₂²×C₄	#1132	1	2	yes	—
175	C₂²×C₄	D₅₆	1	1	yes	characteristic
176	C₂²×C₄	D₅₆	1	1	yes	characteristic
177	C₂²×C₄	Dic₂₈	1	1	yes	characteristic
178	C₂²×C₄	C₄⋊Dic₇	1	1	yes	characteristic
179	C₂²×C₄	C₄⋊Dic₇	1	1	yes	characteristic
180	C₂²×C₄	(C₂×D₇)⋊C₄	1	2	yes	—
181	C₂²×C₄	(C₂×D₇)⋊C₄	1	2	yes	—
182	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
183	M₁₆	—	14	8	no	—
184	M₁₆	—	28	4	no	—
185	M₁₆	—	28	4	no	—
186	M₁₆	—	28	4	no	—
187	C₂×C₈	—	28	4	no	—
188	C₂×C₈	—	28	4	no	—
189	C₂×C₈	—	28	4	no	—
190	C₂×C₈	C₄×Dic₇	1	1	yes	characteristic
191	C₂×C₈	C₄×Dic₇	1	1	yes	characteristic
192	C₂×C₈	C₄×Dic₇	1	2	yes	—
193	C₂×C₈	Dic₇⋊C₄	1	2	yes	—
194	C₂×C₈	Dic₇⋊C₄	1	2	yes	—
195	C₂×C₈	Dic₇⋊C₄	1	2	yes	—
196	C₂×C₈	Dic₇⋊C₄	1	2	yes	—
197	C₂×C₈	C₄⋊Dic₇	1	1	yes	characteristic
198	C₂×C₈	C₄⋊Dic₇	1	1	yes	characteristic
199	C₂×C₈	(C₂×D₇)⋊C₄	1	2	yes	—
200	C₂×C₈	(C₂×D₇)⋊C₄	1	2	yes	—
201	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
202	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
203	C₂×C₈	C₂²⋊Dic₇	1	2	yes	—
204	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
205	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
206	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
207	C₂×C₈	C₂×C₂²⋊D₇	1	2	yes	—
208	C₁₆	—	2	4	no	—
209	C₂×C₇	—	2	2	no	—
210	C₂×C₇	—	2	1	no	—
211	C₂×C₇	—	2	2	no	—
212	C₂×C₇	—	2	1	no	—
213	C₂×C₇	#1385	1	2	yes	complemented
214	C₂×C₇	C₂×#340	1	1	yes	characteristic
215	C₂³	—	4	2	no	—
216	C₂³	#6390	1	2	yes	—
217	C₂³	#6390	1	2	yes	—
218	C₂³	#6392	1	1	yes	characteristic
219	C₂³	#6392	1	1	yes	characteristic
220	C₂³	C₂×Dic₂₈	1	1	yes	characteristic
221	C₂×C₄	#6390	1	2	yes	—
222	C₂×C₄	#6390	1	2	yes	—
223	C₂×C₄	#6391	1	1	yes	characteristic
224	C₂×C₄	#6391	1	2	yes	—
225	C₂×C₄	#6391	1	1	yes	characteristic
226	C₂×C₄	#6392	1	1	yes	characteristic
227	C₂×C₄	#6392	1	1	yes	characteristic
228	C₂×C₄	#6395	1	2	yes	—
229	C₂×C₄	#6395	1	2	yes	—
230	C₂×C₄	#6395	1	2	yes	—
231	C₂×C₄	#6395	1	2	yes	—
232	C₂×C₄	C₇⋊#95	1	2	yes	—
233	C₂×C₄	C₇⋊#95	1	1	yes	characteristic
234	C₂×C₄	C₇⋊#95	1	1	yes	characteristic
235	C₂×C₄	C₂×#1132	1	2	yes	—
236	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
237	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
238	C₂×C₄	C₂×(C₂×D₇)⋊C₄	1	2	yes	—
239	C₈	—	28	8	no	—
240	C₈	C₇⋊#95	1	1	yes	characteristic
241	C₈	C₇⋊#95	1	1	yes	characteristic
242	C₈	C₇⋊#95	1	2	yes	—
243	C₈	C₂×C₄×Dic₇	1	1	yes	characteristic
244	C₈	C₂×Dic₇⋊C₄	1	2	yes	—
245	C₈	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
246	C₇	C₂×#1385	1	1	yes	characteristic, complemented, Sylow
247	C₂²	—	2	4	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	4	1	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	1	no	—
252	C₂²	#89346	1	2	yes	—
253	C₂²	#89346	1	1	yes	characteristic
254	C₂²	#89346	1	1	yes	characteristic
255	C₂²	C₂×#6390	1	2	yes	—
256	C₂²	C₂×#6392	1	1	yes	characteristic
257	C₄	#89346	1	2	yes	—
258	C₄	#89346	1	1	yes	characteristic
259	C₄	#89346	1	1	yes	characteristic
260	C₄	C₂×#6391	1	1	yes	characteristic
261	C₄	C₂×#6395	1	2	yes	—
262	C₄	C₂×C₇⋊#95	1	1	yes	characteristic
263	C₂	—	2	2	no	—
264	C₂	—	2	1	no	—
265	C₂	—	2	2	no	—
266	C₂	—	2	1	no	—
267	C₂	#11725626	1	2	yes	direct product factor
268	C₂	C₂×#89346	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6391
commutator	C₇×C₈	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₂²×C₇×M₃₂	C₂
socle	C₂²×C₇	#340
prime radical	C₂⁴×C₇	Q₁₆
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	C₂²×M₃₂	D₇
hypocenter	C₇	C₂×#1385
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. #11725626	C₂	direct (N × H)
213. C₂×C₇	#1385	semidirect
246. C₇	C₂×#1385	semidirect
267. C₂	#11725626	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×#6393	2
#11725626	2
C₂×#1385	7

Subgroups — C2×#11725626 #13268708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11725626 #13268708