Subgroups — #31789783

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	#22577937	C₂	1	1	yes	maximal, characteristic
2	#22577531	C₂	1	2	yes	maximal, complemented
3	#22577476	C₂	1	2	yes	maximal, complemented
4	#22577474	C₂	1	2	yes	maximal, complemented
5	C₈×C₂²⋊D₁₃₇	C₂	1	2	yes	maximal, complemented
6	#22577465	C₂	1	2	yes	maximal, complemented
7	#22577456	C₂	1	1	yes	maximal, characteristic, complemented
8	#22577360	C₂	1	2	yes	maximal, complemented
9	C₂×#97×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14555493	C₂²	1	2	yes	—
11	C₂×C₈×D₁₃₇	C₂²	1	2	yes	—
12	#17984082	C₄	1	2	yes	—
13	#17984080	C₄	1	1	yes	characteristic
14	#17984076	C₄	1	1	yes	characteristic
15	#17984071	C₄	1	2	yes	—
16	#17984067	C₄	1	1	yes	characteristic
17	#17984062	C₄	1	1	yes	characteristic
18	C₄×Dic₂₇₄	C₂²	1	1	yes	characteristic
19	C₄×D₅₄₈	C₂²	1	1	yes	characteristic
20	C₄×C₂²⋊D₁₃₇	C₂²	1	2	yes	—
21	#17984045	C₂²	1	2	yes	—
22	C₄⋊#11764040	C₂²	1	2	yes	—
23	#17984017	C₂²	1	2	yes	—
24	#17984017	C₂²	1	2	yes	—
25	C₂²⋊#11764040	C₂²	1	1	yes	characteristic
26	C₂²⋊#11764040	C₂²	1	1	yes	characteristic
27	#17984000	C₂²	1	2	yes	—
28	#17984000	C₂²	1	2	yes	—
29	#17983998	C₂²	1	2	yes	—
30	C₄×#11764040	C₂²	1	2	yes	—
31	C₈×Dic₁₃₇	C₂²	1	2	yes	—
32	C₂²×C₈×C₁₃₇	C₂²	1	2	yes	—
33	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
34	#97×C₁₃₇	C₂²	1	1	yes	characteristic
35	#97×C₁₃₇	C₂²	1	2	yes	—
36	#97×C₁₃₇	C₂²	1	1	yes	characteristic
37	C₂×Dic₂₇₄	C₂×C₄	1	1	yes	characteristic
38	C₂×D₅₄₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₂²⋊D₁₃₇	C₂×C₄	1	2	yes	—
40	C₂×C₄×D₁₃₇	C₂³	1	2	yes	—
41	#14555493	—	2	4	no	—
42	C₂×#11764040	C₂³	1	2	yes	—
43	C₂×#11764040	C₂³	1	2	yes	—
44	C₈×D₁₃₇	—	2	4	no	—
45	C₄⋊Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
46	C₄⋊Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
47	#14555489	C₂×C₄	1	2	yes	—
48	#14555489	C₂×C₄	1	2	yes	—
49	#14555489	C₂×C₄	1	2	yes	—
50	#14555489	C₂×C₄	1	2	yes	—
51	C₂²⋊Dic₁₃₇	C₂×C₄	1	2	yes	—
52	#14555487	C₂×C₄	1	2	yes	—
53	#14555487	C₂×C₄	1	2	yes	—
54	#14555487	C₂×C₄	1	2	yes	—
55	#14555487	C₂×C₄	1	2	yes	—
56	C₄×Dic₁₃₇	C₂³	1	2	yes	—
57	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
60	C₂×C₈×C₁₃₇	—	2	4	no	—
61	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
62	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
63	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
64	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
65	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
66	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
67	C₂²×D₁₃₇	C₂²×C₄	1	2	yes	—
68	Dic₂₇₄	—	4	1	no	—
69	D₅₄₈	—	4	1	no	—
70	C₂²⋊D₁₃₇	—	2	4	no	—
71	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
72	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
73	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
74	C₄×D₁₃₇	—	2	2	no	—
75	C₄×D₁₃₇	—	2	2	no	—
76	#11764040	—	2	2	no	—
77	#11764040	—	2	2	no	—
78	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃₇	—	2	2	no	—
80	C₂×C₄×C₁₃₇	—	2	2	no	—
81	C₂×C₄×C₁₃₇	—	4	1	no	—
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	1	yes	characteristic
87	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
88	C₈×C₁₃₇	—	2	2	no	—
89	C₈×C₁₃₇	#41	1	4	yes	—
90	C₂²×C₁₃₇	—	2	1	no	—
91	C₂²×C₁₃₇	—	2	1	no	—
92	C₂²×C₁₃₇	#131	1	2	yes	—
93	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
94	C₂×D₁₃₇	—	2	2	no	—
95	C₂×D₁₃₇	—	2	2	no	—
96	C₂×D₁₃₇	—	2	2	no	—
97	C₄×C₁₃₇	—	2	1	no	—
98	C₄×C₁₃₇	—	2	1	no	—
99	C₄×C₁₃₇	—	2	1	no	—
100	C₄×C₁₃₇	—	2	1	no	—
101	C₄×C₁₃₇	—	2	1	no	—
102	C₄×C₁₃₇	C₂×#41	1	2	yes	—
103	Dic₁₃₇	—	2	2	no	—
104	Dic₁₃₇	—	2	2	no	—
105	Dic₁₃₇	—	2	2	no	—
106	C₂×C₁₃₇	—	2	2	no	—
107	C₂×C₁₃₇	C₄×#41	1	1	yes	characteristic
108	C₂×C₁₃₇	C₂×#131	1	1	yes	characteristic
109	C₂×C₁₃₇	#568	1	1	yes	characteristic
110	D₁₃₇	—	4	2	no	—
111	C₁₃₇	#2968	1	1	yes	characteristic, complemented, Sylow
112	#2968	—	137	1	no	maximal, Sylow
113	#529	—	137	1	no	—
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	D₁₃₇	1	1	yes	characteristic, complemented
122	C₂×M₁₆	—	137	2	no	—
123	C₂²×C₈	—	137	2	no	—
124	C₂²×C₈	C₂×D₁₃₇	1	2	yes	—
125	#126	—	137	2	no	—
126	#125	—	137	1	no	—
127	#124	—	137	1	no	—
128	#123	—	137	2	no	—
129	C₂²⋊Q₈	—	137	1	no	—
130	#121	—	137	1	no	—
131	C₄×Q₈	—	137	1	no	—
132	C₄×D₄	—	137	2	no	—
133	C₄×D₄	—	137	1	no	—
134	#117	—	137	2	no	—
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	2	no	—
144	#97	C₂×D₁₃₇	1	1	yes	characteristic
145	#97	C₂×D₁₃₇	1	1	yes	characteristic
146	#97	C₂×D₁₃₇	1	2	yes	—
147	C₄×C₈	—	137	2	no	—
148	C₄×C₈	—	137	2	no	—
149	C₂×Q₈	—	137	1	no	—
150	C₂×D₄	—	137	2	no	—
151	C₂×D₄	—	137	1	no	—
152	C₂²×C₄	—	137	2	no	—
153	C₂²×C₄	C₄×D₁₃₇	1	1	yes	characteristic
154	C₂²×C₄	C₄×D₁₃₇	1	1	yes	characteristic
155	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
156	M₁₆	—	274	4	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	274	4	no	—
159	C₂×C₈	—	137	2	no	—
160	C₂×C₈	—	137	2	no	—
161	C₂×C₈	C₂²×D₁₃₇	1	2	yes	—
162	C₂×C₈	C₂²×D₁₃₇	1	2	yes	—
163	C₄⋊C₄	—	137	2	no	—
164	C₄⋊C₄	—	137	2	no	—
165	C₄⋊C₄	—	137	2	no	—
166	C₄⋊C₄	—	137	2	no	—
167	C₄⋊C₄	—	137	1	no	—
168	C₄⋊C₄	—	137	1	no	—
169	C₂²⋊C₄	—	137	2	no	—
170	C₂²⋊C₄	—	137	2	no	—
171	C₂²⋊C₄	—	137	2	no	—
172	C₂²⋊C₄	—	137	2	no	—
173	C₂²⋊C₄	—	137	2	no	—
174	C₄²	—	137	2	no	—
175	C₄²	C₄×D₁₃₇	1	1	yes	characteristic
176	C₄²	C₄×D₁₃₇	1	1	yes	characteristic
177	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
178	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
179	C₂³	—	137	2	no	—
180	C₂³	C₂×C₄×D₁₃₇	1	1	yes	characteristic
181	Q₈	—	548	1	no	—
182	D₄	—	274	4	no	—
183	D₄	—	548	1	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	4	1	no	—
187	C₂×C₄	—	274	2	no	—
188	C₂×C₄	—	274	2	no	—
189	C₂×C₄	—	137	2	no	—
190	C₂×C₄	—	137	2	no	—
191	C₂×C₄	—	137	2	no	—
192	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
194	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
195	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
196	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
197	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
198	C₈	—	2	2	no	—
199	C₈	—	274	2	no	—
200	C₈	—	274	2	no	—
201	C₈	#14555510	1	4	yes	—
202	C₂²	—	2	1	no	—
203	C₂²	—	2	1	no	—
204	C₂²	—	274	2	no	—
205	C₂²	—	274	2	no	—
206	C₂²	—	274	2	no	—
207	C₂²	#17984096	1	2	yes	—
208	C₂²	C₂²×C₄×D₁₃₇	1	1	yes	characteristic
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	274	2	no	—
215	C₄	—	274	2	no	—
216	C₄	—	274	2	no	—
217	C₄	C₂×#14555510	1	2	yes	—
218	C₂	—	2	2	no	—
219	C₂	—	548	2	no	—
220	C₂	C₄×#14555510	1	1	yes	characteristic
221	C₂	C₂×#17984096	1	1	yes	characteristic
222	C₂	#22578169	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃₇
commutator, socle	C₂²×C₁₃₇	C₂³×C₄
Fitting	C₂×#97×C₁₃₇	C₂
prime radical	#17984062	C₄
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	C₂×#97	D₁₃₇
hypocenter	C₁₃₇	#2968
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
2. #22577531	C₂	semidirect
3. #22577476	C₂	semidirect
4. #22577474	C₂	semidirect
5. C₈×C₂²⋊D₁₃₇	C₂	semidirect
6. #22577465	C₂	semidirect
7. #22577456	C₂	semidirect
8. #22577360	C₂	semidirect
9. C₂×#97×C₁₃₇	C₂	semidirect
111. C₁₃₇	#2968	semidirect
121. C₂×#97	D₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#22577937	2
#22577531	2
#22577476	2
#22577474	2
C₈×C₂²⋊D₁₃₇	2
#22577465	2
#22577456	2
#22577360	2
C₂×#97×C₁₃₇	2
#2968	137