Subgroups — #31797586

Order 17536.

356 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578455	C₂	1	2	yes	maximal, complemented
2	#22578455	C₂	1	2	yes	maximal, complemented
3	#22578454	C₂	1	2	yes	maximal, complemented
4	#22578452	C₂	1	2	yes	maximal, complemented
5	#22578409	C₂	1	2	yes	maximal
6	C₂×#17984155	C₂	1	1	yes	maximal, characteristic
7	#22578367	C₂	1	2	yes	maximal
8	C₄×#14555519	C₂	1	1	yes	maximal, characteristic
9	#22577955	C₂	1	1	yes	maximal, characteristic
10	C₂×#14555519	C₂²	1	1	yes	characteristic
11	C₂×#14555519	C₂²	1	1	yes	characteristic
12	C₂²×#11764047	C₂²	1	2	yes	—
13	#17984155	C₂²	1	1	yes	characteristic
14	#17984155	C₂²	1	2	yes	—
15	#17984155	C₂²	1	2	yes	—
16	#17984155	C₂²	1	1	yes	characteristic
17	#17984155	C₂²	1	2	yes	—
18	C₄⋊#11764047	C₂²	1	2	yes	—
19	C₄⋊#11764047	C₂²	1	2	yes	—
20	#17984143	C₂²	1	2	yes	—
21	#17984143	C₂²	1	2	yes	—
22	#17984143	C₂²	1	2	yes	—
23	C₄×#11764047	C₂²	1	2	yes	—
24	C₄×#11764047	C₂²	1	2	yes	—
25	C₄×#11764047	C₂²	1	2	yes	—
26	#17984087	C₄	1	1	yes	characteristic
27	#17984086	C₄	1	1	yes	characteristic
28	#17984078	C₄	1	2	yes	—
29	#17984067	C₄	1	1	yes	characteristic
30	#17984065	C₄	1	2	yes	—
31	Q₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
32	#17984053	C₂²	1	2	yes	—
33	#17984053	C₂²	1	1	yes	characteristic
34	#17984044	C₂²	1	2	yes	—
35	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
36	C₂²⋊Q₈×C₁₃₇	C₄	1	1	yes	characteristic
37	#14555519	—	2	2	no	—
38	#14555519	Q₈	1	4	yes	complemented
39	C₂×#11764047	—	2	4	no	—
40	C₂×#11764047	C₂³	1	2	yes	—
41	C₂×#11764047	C₂³	1	2	yes	—
42	C₂×#11764047	C₂³	1	2	yes	—
43	C₂×#11764047	C₂³	1	2	yes	—
44	C₂×Dic₂₇₄	C₂×C₄	1	2	yes	—
45	C₂×Dic₂₇₄	C₂×C₄	1	1	yes	characteristic
46	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
47	C₂²×Dic₁₃₇	C₂³	1	1	yes	characteristic
48	C₄⋊Dic₁₃₇	C₂×C₄	1	2	yes	—
49	C₄⋊Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
50	#14555489	C₂×C₄	1	2	yes	—
51	#14555489	C₂×C₄	1	2	yes	—
52	#14555489	C₂×C₄	1	2	yes	—
53	#14555489	C₂×C₄	1	2	yes	—
54	#14555489	C₂×C₄	1	2	yes	—
55	C₂²⋊Dic₁₃₇	C₂×C₄	1	2	yes	—
56	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
57	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
58	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
59	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
60	C₄×Dic₁₃₇	C₂³	1	2	yes	—
61	C₄×Dic₁₃₇	C₂³	1	2	yes	—
62	C₂×Q₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
64	C₄⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
65	C₄⋊C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
66	C₂²⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
67	#11764047	—	2	2	no	—
68	#11764047	—	2	2	no	—
69	#11764047	—	2	2	no	—
70	#11764047	C₂×Q₈	1	4	yes	—
71	Dic₂₇₄	—	2	4	no	—
72	Dic₂₇₄	—	4	1	no	—
73	C₂×Dic₁₃₇	—	2	2	no	—
74	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
75	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
76	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
77	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×Dic₁₃₇	C₂×Q₈	1	2	yes	—
80	C₂×Dic₁₃₇	#41	1	2	yes	—
81	C₂×Dic₁₃₇	C₂⁴	1	1	yes	characteristic
82	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
83	Q₈×C₁₃₇	—	4	1	no	—
84	C₂×C₄×C₁₃₇	—	4	1	no	—
85	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
86	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₁₃₇	—	2	1	no	—
90	C₂²×C₁₃₇	—	2	1	no	—
91	C₂²×C₁₃₇	C₄×Q₈	1	2	yes	—
92	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
93	C₄×C₁₃₇	—	2	2	no	—
94	C₄×C₁₃₇	—	2	1	no	—
95	C₄×C₁₃₇	—	2	1	no	—
96	C₄×C₁₃₇	—	2	2	no	—
97	Dic₁₃₇	—	2	1	no	—
98	Dic₁₃₇	—	2	2	no	—
99	Dic₁₃₇	—	2	2	no	—
100	Dic₁₃₇	C₄×Q₈	1	2	yes	—
101	Dic₁₃₇	#131	1	2	yes	—
102	Dic₁₃₇	C₂²×Q₈	1	1	yes	characteristic
103	Dic₁₃₇	C₂×#41	1	1	yes	characteristic
104	C₂×C₁₃₇	—	2	2	no	—
105	C₂×C₁₃₇	C₂×C₄×Q₈	1	1	yes	characteristic
106	C₂×C₁₃₇	C₂×#131	1	1	yes	characteristic
107	C₂×C₁₃₇	#568	1	1	yes	characteristic
108	C₁₃₇	#2962	1	1	yes	characteristic, complemented, Sylow
109	#2962	—	137	1	no	maximal, Sylow
110	#533	—	137	1	no	—
111	#446	—	137	2	no	—
112	#446	—	137	2	no	—
113	#446	—	137	2	no	—
114	C₈×Q₈	—	137	2	no	—
115	#424	—	137	2	no	—
116	C₂×C₄⋊C₈	—	137	1	no	—
117	#405	—	137	2	no	—
118	C₄×M₁₆	—	137	1	no	—
119	C₂×M₁₆	—	137	1	no	—
120	C₂×M₁₆	—	137	1	no	—
121	C₂²×C₈	—	137	2	no	—
122	C₄⋊Q₈	—	137	1	no	—
123	C₄⋊Q₈	—	137	1	no	—
124	#125	—	137	2	no	—
125	C₂²⋊Q₈	—	137	1	no	—
126	C₂²⋊Q₈	—	137	2	no	—
127	C₂²⋊Q₈	#10587739	1	1	yes	characteristic
128	C₄×Q₈	—	137	1	no	—
129	C₄×Q₈	—	137	2	no	—
130	C₄×Q₈	—	137	1	no	—
131	#117	—	137	2	no	—
132	C₂×C₄²	—	137	1	no	—
133	C₄⋊C₈	—	137	2	no	—
134	C₄⋊C₈	—	137	2	no	—
135	C₄⋊C₈	—	137	1	no	—
136	C₄⋊C₈	—	137	1	no	—
137	C₄⋊C₈	—	137	2	no	—
138	C₄⋊C₈	—	137	2	no	—
139	C₄⋊C₈	—	137	2	no	—
140	#97	—	137	2	no	—
141	#97	—	137	2	no	—
142	#97	—	137	2	no	—
143	C₄×C₈	—	137	2	no	—
144	C₄×C₈	—	137	2	no	—
145	C₄×C₈	—	137	2	no	—
146	C₂×Q₈	—	137	2	no	—
147	C₂×Q₈	—	137	1	no	—
148	C₂×Q₈	C₂×#10587739	1	1	yes	characteristic
149	C₂²×C₄	—	137	1	no	—
150	C₂²×C₄	—	137	1	no	—
151	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
152	M₁₆	—	274	2	no	—
153	M₁₆	—	137	4	no	—
154	C₂×C₈	—	274	4	no	—
155	C₂×C₈	—	137	2	no	—
156	C₂×C₈	—	137	2	no	—
157	C₂×C₈	—	137	2	no	—
158	C₂×C₈	—	137	2	no	—
159	C₄⋊C₄	—	137	2	no	—
160	C₄⋊C₄	—	137	2	no	—
161	C₄⋊C₄	—	137	2	no	—
162	C₄⋊C₄	—	137	2	no	—
163	C₄⋊C₄	—	137	2	no	—
164	C₄⋊C₄	—	137	2	no	—
165	C₄⋊C₄	—	137	1	no	—
166	C₄⋊C₄	C₂×#10587739	1	2	yes	—
167	C₄⋊C₄	C₂×#10587739	1	1	yes	characteristic
168	C₂²⋊C₄	—	137	2	no	—
169	C₂²⋊C₄	C₂×#10587739	1	2	yes	—
170	C₄²	—	137	1	no	—
171	C₄²	—	137	1	no	—
172	C₄²	—	137	1	no	—
173	C₄²	—	137	2	no	—
174	C₄²	—	137	2	no	—
175	C₄²	—	137	1	no	—
176	C₂³	C₂²×#10587739	1	1	yes	characteristic
177	Q₈	—	4	1	no	—
178	Q₈	—	274	4	no	—
179	Q₈	—	548	1	no	—
180	C₂×C₄	—	4	1	no	—
181	C₂×C₄	—	274	2	no	—
182	C₂×C₄	—	137	1	no	—
183	C₂×C₄	—	137	2	no	—
184	C₂×C₄	—	137	2	no	—
185	C₂×C₄	—	137	2	no	—
186	C₂×C₄	—	137	2	no	—
187	C₂×C₄	—	137	1	no	—
188	C₂×C₄	—	137	1	no	—
189	C₂×C₄	—	137	1	no	—
190	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
191	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
192	C₂×C₄	C₂²×#10587739	1	2	yes	—
193	C₂×C₄	C₂²×#10587739	1	2	yes	—
194	C₈	—	274	2	no	—
195	C₈	—	274	2	no	—
196	C₈	—	274	2	no	—
197	C₈	—	137	4	no	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	Q₈×#10587739	1	2	yes	—
201	C₂²	C₂³×#10587739	1	1	yes	characteristic
202	C₄	—	2	2	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	2	2	no	—
206	C₄	—	274	1	no	—
207	C₄	—	274	2	no	—
208	C₄	—	274	2	no	—
209	C₄	—	137	1	no	—
210	C₄	—	137	1	no	—
211	C₄	—	137	2	no	—
212	C₄	—	137	2	no	—
213	C₂	—	2	2	no	—
214	C₂	C₂×Q₈×#10587739	1	1	yes	characteristic
215	C₂	D₁₃₇⋊#131	1	1	yes	characteristic
216	C₂	#22578481	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#10587739
commutator, socle	C₂²×C₁₃₇	C₂³×C₄
Fitting	C₂²⋊Q₈×C₁₃₇	C₄
prime radical	C₂³×C₁₃₇	C₂²×C₄
cosocle	C₂×Dic₁₃₇	C₂⁴
hypercenter	C₂²⋊Q₈	#10587739
hypocenter	C₁₃₇	#2962
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #22578455	C₂	semidirect
2. #22578455	C₂	semidirect
3. #22578454	C₂	semidirect
4. #22578452	C₂	semidirect
38. #14555519	Q₈	semidirect
108. C₁₃₇	#2962	semidirect

Maximal subgroups

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

subgroup	index
#22578455	2
#22578454	2
#22578452	2
#22578409	2
C₂×#17984155	2
#22578367	2
C₄×#14555519	2
#22577955	2
#2962	137