Subgroups — #31789785

Order 17536.

372 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22577534	C₂	1	2	yes	maximal, complemented
2	#22577475	C₂	1	2	yes	maximal, complemented
3	#22577475	C₂	1	2	yes	maximal, complemented
4	D₄×#11764040	C₂	1	2	yes	maximal, complemented
5	#22577464	C₂	1	2	yes	maximal, complemented
6	#22577456	C₂	1	1	yes	maximal, characteristic, complemented
7	#22577359	C₂	1	2	yes	maximal, complemented
8	C₂×#17983999	C₂	1	1	yes	maximal, characteristic, complemented
9	#529×C₁₃₇	C₂	1	1	yes	maximal, characteristic
10	C₂×#14555494	C₂²	1	2	yes	—
11	C₂²×#11764040	C₂²	1	2	yes	—
12	C₂²×#11764040	C₂²	1	2	yes	—
13	C₄⋊#11764040	C₂²	1	2	yes	—
14	C₄⋊#11764040	C₂²	1	2	yes	—
15	C₄⋊#11764040	C₂²	1	2	yes	—
16	C₂²⋊#11764040	C₂²	1	1	yes	characteristic
17	C₂²⋊#11764040	C₂²	1	1	yes	characteristic
18	C₂²⋊#11764040	C₂²	1	2	yes	—
19	C₂²⋊#11764040	C₂²	1	2	yes	—
20	#17983999	C₂²	1	1	yes	characteristic
21	#17983999	C₂²	1	2	yes	—
22	#17983999	C₂²	1	2	yes	—
23	#17983999	C₂²	1	1	yes	characteristic
24	C₄×#11764040	C₂²	1	2	yes	—
25	C₄×#11764040	C₂²	1	2	yes	—
26	#126×C₁₃₇	C₄	1	2	yes	—
27	#125×C₁₃₇	C₄	1	1	yes	characteristic
28	#124×C₁₃₇	C₄	1	1	yes	characteristic
29	#123×C₁₃₇	C₄	1	2	yes	—
30	C₂²⋊Q₈×C₁₃₇	C₄	1	1	yes	characteristic
31	#121×C₁₃₇	C₄	1	1	yes	characteristic
32	C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
33	C₄×D₄×C₁₃₇	C₂²	1	2	yes	—
34	C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
35	#117×C₁₃₇	C₂²	1	2	yes	—
36	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
37	#14555494	—	2	4	no	—
38	C₂×#11764040	—	2	4	no	—
39	C₂×#11764040	—	2	4	no	—
40	C₂×#11764040	C₂³	1	2	yes	—
41	C₂×#11764040	C₂³	1	2	yes	—
42	C₂×#11764040	C₂³	1	2	yes	—
43	C₂×#11764040	C₂³	1	2	yes	—
44	C₂×Q₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
45	C₂×D₄×C₁₃₇	C₂×C₄	1	2	yes	—
46	C₂×D₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
51	C₄⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
52	C₄⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
53	C₄⋊C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
54	C₄⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
55	C₄⋊C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
56	C₄⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
57	C₂²⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
58	C₂²⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
59	C₂²⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
60	C₂²⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
61	C₂²⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
62	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
63	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
64	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
65	C₄²×C₁₃₇	C₂³	1	2	yes	—
66	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
67	#11764040	—	2	2	no	—
68	#11764040	—	2	2	no	—
69	#11764040	—	2	2	no	—
70	#11764040	#41	1	4	yes	—
71	C₂³×C₁₃₇	C₂²×C₄	1	2	yes	—
72	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
73	Q₈×C₁₃₇	—	4	1	no	—
74	D₄×C₁₃₇	—	2	4	no	—
75	D₄×C₁₃₇	—	4	1	no	—
76	C₂×C₄×C₁₃₇	—	2	2	no	—
77	C₂×C₄×C₁₃₇	—	2	2	no	—
78	C₂×C₄×C₁₃₇	—	2	2	no	—
79	C₂×C₄×C₁₃₇	—	2	2	no	—
80	C₂×C₄×C₁₃₇	—	4	1	no	—
81	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
90	C₂²×C₁₃₇	—	2	1	no	—
91	C₂²×C₁₃₇	—	2	1	no	—
92	C₂²×C₁₃₇	—	2	2	no	—
93	C₂²×C₁₃₇	—	2	2	no	—
94	C₂²×C₁₃₇	—	2	2	no	—
95	C₂²×C₁₃₇	#131	1	2	yes	—
96	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
97	C₄×C₁₃₇	—	2	1	no	—
98	C₄×C₁₃₇	—	2	2	no	—
99	C₄×C₁₃₇	—	2	1	no	—
100	C₄×C₁₃₇	—	2	1	no	—
101	C₄×C₁₃₇	—	2	2	no	—
102	C₄×C₁₃₇	—	2	1	no	—
103	C₄×C₁₃₇	—	2	1	no	—
104	C₄×C₁₃₇	—	2	2	no	—
105	C₄×C₁₃₇	C₂×#41	1	2	yes	—
106	C₂×C₁₃₇	—	2	2	no	—
107	C₂×C₁₃₇	—	4	2	no	—
108	C₂×C₁₃₇	C₄×#41	1	1	yes	characteristic
109	C₂×C₁₃₇	C₂×#131	1	1	yes	characteristic
110	C₂×C₁₃₇	#568	1	1	yes	characteristic
111	C₁₃₇	#2968	1	1	yes	characteristic, complemented, Sylow
112	#2968	—	137	1	no	maximal, Sylow
113	#529	D₁₃₇	1	1	yes	characteristic
114	#446	—	137	2	no	—
115	#435	—	137	2	no	—
116	#435	—	137	2	no	—
117	C₈×D₄	—	137	2	no	—
118	#433	—	137	2	no	—
119	#431	—	137	1	no	—
120	#405	—	137	2	no	—
121	C₂×#97	—	137	1	no	—
122	C₂×M₁₆	—	137	2	no	—
123	C₂²×C₈	—	137	2	no	—
124	C₂²×C₈	—	137	2	no	—
125	#126	Dic₁₃₇	1	2	yes	—
126	#125	Dic₁₃₇	1	1	yes	characteristic
127	#124	Dic₁₃₇	1	1	yes	characteristic
128	#123	Dic₁₃₇	1	2	yes	—
129	C₂²⋊Q₈	Dic₁₃₇	1	1	yes	characteristic
130	#121	Dic₁₃₇	1	1	yes	characteristic
131	C₄×Q₈	C₂×D₁₃₇	1	1	yes	characteristic
132	C₄×D₄	C₂×D₁₃₇	1	2	yes	—
133	C₄×D₄	C₂×D₁₃₇	1	1	yes	characteristic
134	#117	C₂×D₁₃₇	1	2	yes	—
135	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
136	C₄⋊C₈	—	137	2	no	—
137	C₄⋊C₈	—	137	2	no	—
138	C₄⋊C₈	—	137	2	no	—
139	C₂²⋊C₈	—	137	1	no	—
140	C₂²⋊C₈	—	137	1	no	—
141	C₂²⋊C₈	—	137	2	no	—
142	C₂²⋊C₈	—	137	2	no	—
143	#97	—	137	1	no	—
144	#97	—	137	1	no	—
145	#97	—	137	2	no	—
146	#97	—	137	2	no	—
147	C₄×C₈	—	137	2	no	—
148	C₄×C₈	—	137	2	no	—
149	C₂×Q₈	C₂×Dic₁₃₇	1	1	yes	characteristic
150	C₂×D₄	C₂×Dic₁₃₇	1	2	yes	—
151	C₂×D₄	C₂×Dic₁₃₇	1	1	yes	characteristic
152	C₂²×C₄	C₂×Dic₁₃₇	1	1	yes	characteristic
153	C₂²×C₄	C₂×Dic₁₃₇	1	1	yes	characteristic
154	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
155	C₂²×C₄	C₂²×D₁₃₇	1	2	yes	—
156	M₁₆	—	274	4	no	—
157	C₂×C₈	—	274	4	no	—
158	C₂×C₈	—	274	4	no	—
159	C₂×C₈	—	137	2	no	—
160	C₂×C₈	—	137	2	no	—
161	C₂×C₈	—	137	2	no	—
162	C₂×C₈	—	137	2	no	—
163	C₄⋊C₄	C₂×Dic₁₃₇	1	2	yes	—
164	C₄⋊C₄	C₂×Dic₁₃₇	1	2	yes	—
165	C₄⋊C₄	C₂×Dic₁₃₇	1	1	yes	characteristic
166	C₄⋊C₄	C₂×Dic₁₃₇	1	2	yes	—
167	C₄⋊C₄	C₂×Dic₁₃₇	1	1	yes	characteristic
168	C₄⋊C₄	C₂×Dic₁₃₇	1	2	yes	—
169	C₂²⋊C₄	C₂×Dic₁₃₇	1	2	yes	—
170	C₂²⋊C₄	C₂×Dic₁₃₇	1	2	yes	—
171	C₂²⋊C₄	C₂×Dic₁₃₇	1	2	yes	—
172	C₂²⋊C₄	C₂×Dic₁₃₇	1	2	yes	—
173	C₂²⋊C₄	C₂×Dic₁₃₇	1	2	yes	—
174	C₄²	C₂×Dic₁₃₇	1	1	yes	characteristic
175	C₄²	C₂×Dic₁₃₇	1	1	yes	characteristic
176	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
177	C₄²	C₂²×D₁₃₇	1	2	yes	—
178	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
179	C₂³	C₂²×Dic₁₃₇	1	2	yes	—
180	C₂³	C₂²×Dic₁₃₇	1	1	yes	characteristic
181	Q₈	—	4	1	no	—
182	D₄	—	2	4	no	—
183	D₄	—	4	1	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	4	1	no	—
189	C₂×C₄	C₂²×Dic₁₃₇	1	1	yes	characteristic
190	C₂×C₄	C₂²×Dic₁₃₇	1	1	yes	characteristic
191	C₂×C₄	C₂²×Dic₁₃₇	1	1	yes	characteristic
192	C₂×C₄	C₂²×Dic₁₃₇	1	1	yes	characteristic
193	C₂×C₄	C₂²×Dic₁₃₇	1	1	yes	characteristic
194	C₂×C₄	C₂²×Dic₁₃₇	1	2	yes	—
195	C₂×C₄	C₂²×Dic₁₃₇	1	2	yes	—
196	C₂×C₄	C₂²×Dic₁₃₇	1	2	yes	—
197	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
198	C₈	—	274	2	no	—
199	C₈	—	274	2	no	—
200	C₈	—	274	2	no	—
201	C₈	—	137	4	no	—
202	C₂²	—	2	1	no	—
203	C₂²	—	2	1	no	—
204	C₂²	—	2	2	no	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	#17984095	1	2	yes	—
208	C₂²	C₂³×Dic₁₃₇	1	1	yes	characteristic
209	C₄	—	2	1	no	—
210	C₄	—	2	2	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	2	no	—
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	2	no	—
217	C₄	#41×D₁₃₇	1	2	yes	—
218	C₂	—	2	2	no	—
219	C₂	—	4	2	no	—
220	C₂	#41×Dic₁₃₇	1	1	yes	characteristic
221	C₂	C₂×#17984095	1	1	yes	characteristic
222	C₂	#22578170	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃₇
commutator, socle	C₂²×C₁₃₇	C₂³×C₄
Fitting	#529×C₁₃₇	C₂
prime radical	#121×C₁₃₇	C₄
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	#529	D₁₃₇
hypocenter	C₁₃₇	#2968
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ #529

Complements

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

N	H ≅ G/N	type
1. #22577534	C₂	semidirect
2. #22577475	C₂	semidirect
3. #22577475	C₂	semidirect
4. D₄×#11764040	C₂	semidirect
5. #22577464	C₂	semidirect
6. #22577456	C₂	semidirect
7. #22577359	C₂	semidirect
8. C₂×#17983999	C₂	semidirect
111. C₁₃₇	#2968	semidirect

Maximal subgroups

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

subgroup	index
#22577534	2
#22577475	2
D₄×#11764040	2
#22577464	2
#22577456	2
#22577359	2
C₂×#17983999	2
#529×C₁₃₇	2
#2968	137