Subgroups — #13509333

Order 1792.

602 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11735915	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×C₇×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#11729167	C₂	1	4	yes	maximal, complemented
4	#11726814	C₂	1	1	yes	maximal, characteristic, complemented
5	#11726812	C₂	1	2	yes	maximal, complemented
6	C₄×D₁₁₂	C₂	1	4	yes	maximal, complemented
7	#11726798	C₂	1	1	yes	maximal, characteristic, complemented
8	#90168	—	2	2	no	—
9	C₂²×C₇×C₁₆	C₂²	1	2	yes	—
10	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
11	#89887	—	2	2	no	—
12	#89887	C₂²	1	4	yes	—
13	C₂×D₁₁₂	C₂²	1	4	yes	—
14	#89467	C₂²	1	2	yes	—
15	#89466	—	2	2	no	—
16	C₄×D₅₆	—	2	2	no	—
17	C₄×D₅₆	C₂²	1	4	yes	—
18	#89454	C₂²	1	2	yes	—
19	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
20	C₄×C₇×C₁₆	C₂²	1	2	yes	—
21	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
22	#89304	C₂²	1	8	yes	—
23	#89299	C₂²	1	4	yes	—
24	#33733	—	7	1	no	maximal, Sylow
25	C₂×#1157	—	4	2	no	—
26	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
27	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
29	#6493	—	2	4	no	—
30	#6493	—	4	2	no	—
31	C₂×D₅₆	—	2	2	no	—
32	C₂×D₅₆	C₂³	1	4	yes	—
33	#6438	—	4	2	no	—
34	#6437	—	2	2	no	—
35	C₄×D₂₈	—	2	4	no	—
36	C₄×D₂₈	—	4	2	no	—
37	#6432	—	2	2	no	—
38	C₂×C₇×C₁₆	—	2	4	no	—
39	C₂×C₇×C₁₆	C₂³	1	2	yes	—
40	C₂×C₇×C₁₆	C₂³	1	2	yes	—
41	C₄×C₇×C₈	D₄	1	1	yes	characteristic
42	C₄×C₇×C₈	D₄	1	1	yes	characteristic
43	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
44	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
45	#6395	—	2	4	no	—
46	#6395	—	2	4	no	—
47	#6392	—	2	2	no	—
48	#6392	C₂³	1	4	yes	—
49	D₁₁₂	—	2	8	no	—
50	#3169	—	7	2	no	—
51	#2252	—	7	1	no	—
52	#2246	—	7	1	no	—
53	#2240	—	7	2	no	—
54	#2215	—	7	4	no	—
55	C₄×D₁₆	—	7	4	no	—
56	C₂×C₄×C₁₆	D₇	1	1	yes	characteristic, complemented
57	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₂²⋊D₇	—	4	4	no	—
61	#1157	—	8	8	no	—
62	C₂×D₂₈	—	2	4	no	—
63	C₂×D₂₈	—	4	2	no	—
64	C₂×D₂₈	—	4	2	no	—
65	C₂×C₄×D₇	—	4	4	no	—
66	C₂×Dic₁₄	—	4	2	no	—
67	C₂×C₇×C₈	—	2	2	no	—
68	C₂×C₇×C₈	—	2	2	no	—
69	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
70	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
71	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
72	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
73	C₄²×C₇	D₈	1	1	yes	characteristic
74	C₄²×C₇	D₈	1	1	yes	characteristic
75	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
77	(C₂×D₇)⋊C₄	—	4	4	no	—
78	(C₂×D₇)⋊C₄	—	4	4	no	—
79	C₄⋊Dic₇	—	2	4	no	—
80	C₄⋊Dic₇	—	4	2	no	—
81	D₅₆	—	2	8	no	—
82	D₅₆	—	4	4	no	—
83	C₇×C₁₆	—	2	2	no	—
84	C₇×C₁₆	#41	1	4	yes	complemented
85	#532	—	14	2	no	—
86	C₂×D₁₆	—	7	4	no	—
87	C₂²×C₁₆	C₂×D₇	1	2	yes	—
88	#500	—	7	2	no	—
89	#493	—	7	2	no	—
90	#486	—	14	2	no	—
91	#466	—	7	4	no	—
92	#466	—	14	2	no	—
93	C₄×D₈	—	7	4	no	—
94	C₄×D₈	—	14	2	no	—
95	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
96	#366	—	7	4	no	—
97	#357	—	7	8	no	—
98	C₄×C₁₆	C₂×D₇	1	2	yes	—
99	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
100	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
101	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
102	C₂²×D₇	—	4	4	no	—
103	C₂×C₄×C₇	—	2	1	no	—
104	C₂×C₄×C₇	—	2	1	no	—
105	C₂×C₄×C₇	—	2	2	no	—
106	C₂×C₄×C₇	D₁₆	1	4	yes	complemented
107	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
109	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
110	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
113	C₂²⋊D₇	—	8	8	no	—
114	C₂×Dic₇	—	4	4	no	—
115	D₂₈	—	4	4	no	—
116	D₂₈	—	4	4	no	—
117	D₂₈	—	8	4	no	—
118	D₂₈	—	8	4	no	—
119	C₄×D₇	—	8	8	no	—
120	Dic₁₄	—	8	4	no	—
121	C₇×C₈	—	2	1	no	—
122	C₇×C₈	—	2	1	no	—
123	C₇×C₈	#135	1	2	yes	—
124	C₇×C₈	C₂×#41	1	2	yes	—
125	C₂×#41	—	28	2	no	—
126	C₂×D₈	—	7	4	no	—
127	C₂×D₈	—	14	2	no	—
128	C₂²×C₈	D₂₈	1	1	yes	characteristic
129	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
130	C₄⋊Q₈	—	14	2	no	—
131	#127	—	14	2	no	—
132	#124	—	28	2	no	—
133	#121	—	14	4	no	—
134	#121	—	28	2	no	—
135	C₄×D₄	—	14	4	no	—
136	C₄×D₄	—	28	2	no	—
137	C₂×C₄²	D₂₈	1	1	yes	characteristic
138	D₁₆	—	14	8	no	—
139	C₂×C₁₆	—	2	4	no	—
140	C₂×C₁₆	C₂²×D₇	1	2	yes	—
141	C₂×C₁₆	C₂²×D₇	1	2	yes	—
142	#107	—	7	4	no	—
143	#107	—	14	2	no	—
144	#102	—	14	4	no	—
145	#102	—	14	4	no	—
146	C₄×C₈	D₂₈	1	1	yes	characteristic
147	C₄×C₈	D₂₈	1	1	yes	characteristic
148	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
149	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
150	C₂²×C₇	—	2	1	no	—
151	C₂²×C₇	—	2	1	no	—
152	C₂²×C₇	C₂×D₁₆	1	2	yes	—
153	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
154	C₂×D₇	—	8	4	no	—
155	C₂×D₇	—	8	4	no	—
156	C₂×D₇	—	8	4	no	—
157	C₄×C₇	—	2	1	no	—
158	C₄×C₇	—	2	1	no	—
159	C₄×C₇	C₂×D₁₆	1	2	yes	—
160	C₄×C₇	C₂×D₁₆	1	2	yes	—
161	C₄×C₇	#508	1	2	yes	—
162	C₄×C₇	#532	1	1	yes	characteristic
163	C₄×C₇	C₂×#135	1	1	yes	characteristic
164	Dic₇	—	8	4	no	—
165	#41	—	56	8	no	—
166	C₂×Q₈	—	28	2	no	—
167	C₂×D₄	—	14	4	no	—
168	C₂×D₄	—	28	4	no	—
169	C₂×D₄	—	28	2	no	—
170	C₂×D₄	—	28	2	no	—
171	C₂²×C₄	—	28	4	no	—
172	C₂²×C₄	D₅₆	1	1	yes	characteristic
173	C₂²×C₄	D₅₆	1	1	yes	characteristic
174	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
175	D₈	—	14	8	no	—
176	D₈	—	28	4	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
180	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
181	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
182	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
183	C₄⋊C₄	—	14	4	no	—
184	C₄⋊C₄	—	28	2	no	—
185	C₂²⋊C₄	—	28	4	no	—
186	C₂²⋊C₄	—	28	4	no	—
187	C₄²	D₅₆	1	1	yes	characteristic
188	C₄²	D₅₆	1	1	yes	characteristic
189	C₄²	C₂×D₂₈	1	1	yes	characteristic
190	C₄²	C₂×D₂₈	1	1	yes	characteristic
191	C₁₆	—	2	2	no	—
192	C₁₆	#1157	1	4	yes	complemented
193	C₂×C₇	—	2	2	no	—
194	C₂×C₇	#3169	1	1	yes	characteristic
195	C₂×C₇	C₂²×D₁₆	1	1	yes	characteristic
196	C₂×C₇	C₂×#508	1	1	yes	characteristic
197	D₇	—	16	4	no	—
198	C₂³	—	28	4	no	—
199	C₂³	C₂×D₅₆	1	1	yes	characteristic
200	Q₈	—	56	4	no	—
201	D₄	—	28	4	no	—
202	D₄	—	28	4	no	—
203	D₄	—	56	8	no	—
204	D₄	—	56	4	no	—
205	D₄	—	56	4	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	28	4	no	—
210	C₂×C₄	—	56	8	no	—
211	C₂×C₄	D₁₁₂	1	4	yes	complemented
212	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
213	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
214	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
215	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
216	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
217	C₂×C₄	C₂²×D₂₈	1	1	yes	characteristic
218	C₈	—	2	1	no	—
219	C₈	—	2	1	no	—
220	C₈	#6467	1	2	yes	—
221	C₈	C₂×#1157	1	2	yes	—
222	C₇	#33733	1	1	yes	characteristic, complemented, Sylow
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	56	4	no	—
226	C₂²	—	56	4	no	—
227	C₂²	—	56	4	no	—
228	C₂²	C₂×D₁₁₂	1	2	yes	—
229	C₂²	C₂²×D₅₆	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	56	4	no	—
233	C₄	C₂×D₁₁₂	1	2	yes	—
234	C₄	C₂×D₁₁₂	1	2	yes	—
235	C₄	#89676	1	2	yes	—
236	C₄	#90168	1	1	yes	characteristic
237	C₄	C₂×#6467	1	1	yes	characteristic
238	C₂	—	2	2	no	—
239	C₂	—	112	4	no	—
240	C₂	#11735915	1	1	yes	characteristic
241	C₂	C₂²×D₁₁₂	1	1	yes	characteristic
242	C₂	C₂×#89676	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₅₆
commutator, cosocle	C₂×C₇×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₂×C₄×C₇×C₁₆	C₂
socle	C₂²×C₇	C₂²×D₈
hypercenter	C₂×C₄×C₁₆	D₇
hypocenter	C₇	#33733
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. #11735915	C₂	semidirect
2. C₂×C₄×C₇×C₁₆	C₂	semidirect
3. #11729167	C₂	semidirect
4. #11726814	C₂	semidirect
5. #11726812	C₂	semidirect
6. C₄×D₁₁₂	C₂	semidirect
7. #11726798	C₂	semidirect
56. C₂×C₄×C₁₆	D₇	semidirect
84. C₇×C₁₆	#41	semidirect
106. C₂×C₄×C₇	D₁₆	semidirect
192. C₁₆	#1157	semidirect
211. C₂×C₄	D₁₁₂	semidirect
222. C₇	#33733	semidirect

Maximal subgroups

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

subgroup	index
#11735915	2
C₂×C₄×C₇×C₁₆	2
#11729167	2
#11726814	2
#11726812	2
C₄×D₁₁₂	2
#11726798	2
#33733	7