Subgroups — #31797740

Order 17536.

364 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	#22578454	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×#11764047	C₂	1	1	yes	maximal, characteristic, complemented
3	#22578437	C₂	1	1	yes	maximal, characteristic, complemented
4	#22578432	C₂	1	2	yes	maximal, complemented
5	#22578431	C₂	1	1	yes	maximal, characteristic, complemented
6	#22578430	C₂	1	2	yes	maximal, complemented
7	#22578428	C₂	1	2	yes	maximal, complemented
8	#22578420	C₂	1	1	yes	maximal, characteristic, complemented
9	#22578410	C₂	1	2	yes	maximal, complemented
10	C₂×C₄⋊#11764047	C₂	1	1	yes	maximal, characteristic, complemented
11	#22577939	C₂	1	1	yes	maximal, characteristic
12	C₂×#14555520	C₂²	1	2	yes	—
13	C₂×#14555518	C₂²	1	2	yes	—
14	C₂²×#11764047	C₂²	1	2	yes	—
15	#17984155	C₂²	1	2	yes	—
16	#17984155	C₂²	1	1	yes	characteristic
17	#17984155	C₂²	1	1	yes	characteristic
18	C₄⋊#11764047	C₂²	1	2	yes	—
19	C₄⋊#11764047	C₂²	1	1	yes	characteristic
20	C₄⋊#11764047	C₂²	1	1	yes	characteristic
21	C₄⋊#11764047	C₂²	1	2	yes	—
22	#17984145	C₂²	1	2	yes	—
23	#17984145	C₂²	1	2	yes	—
24	C₂²⋊#11764047	C₂²	1	1	yes	characteristic
25	C₂²⋊#11764047	C₂²	1	1	yes	characteristic
26	#17984143	C₂²	1	2	yes	—
27	C₄×#11764047	C₂²	1	1	yes	characteristic
28	C₄×#11764047	C₂²	1	2	yes	—
29	C₄×#11764047	C₂²	1	1	yes	characteristic
30	#17984083	C₄	1	2	yes	—
31	#17984079	C₄	1	1	yes	characteristic
32	#17984077	C₄	1	1	yes	characteristic
33	#17984072	C₄	1	2	yes	—
34	#17984062	C₄	1	1	yes	characteristic
35	Q₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
36	#17984050	C₂²	1	1	yes	characteristic
37	#17984048	C₂²	1	2	yes	—
38	#17984046	C₂²	1	2	yes	—
39	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
40	C₂²⋊Q₈×C₁₃₇	C₄	1	1	yes	characteristic
41	#14555520	—	4	2	no	—
42	#14555518	—	4	2	no	—
43	C₂×#11764047	—	4	2	no	—
44	C₂×#11764047	C₂³	1	1	yes	characteristic
45	C₂×#11764047	C₂³	1	1	yes	characteristic
46	C₂×#11764047	C₂³	1	1	yes	characteristic
47	C₂×#11764047	C₂³	1	1	yes	characteristic
48	C₂×#11764047	C₂³	1	2	yes	—
49	C₂×#11764047	C₂³	1	2	yes	—
50	C₂×D₅₄₈	C₂×C₄	1	1	yes	characteristic
51	C₂×C₂²⋊D₁₃₇	C₂×C₄	1	2	yes	—
52	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
53	C₂²×Dic₁₃₇	C₂³	1	1	yes	characteristic
54	C₂×C₄×D₁₃₇	C₂³	1	2	yes	—
55	C₄⋊Dic₁₃₇	C₂×C₄	1	2	yes	—
56	C₄⋊Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
57	C₄⋊Dic₁₃₇	C₂×C₄	1	2	yes	—
58	#14555489	C₂×C₄	1	2	yes	—
59	#14555487	C₂×C₄	1	2	yes	—
60	#14555487	C₂×C₄	1	2	yes	—
61	#14555487	C₂×C₄	1	2	yes	—
62	#14555487	C₂×C₄	1	2	yes	—
63	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
66	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
67	C₄×Dic₁₃₇	C₂³	1	2	yes	—
68	C₂×Q₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
70	C₄⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
71	C₄⋊C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
72	C₂²⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
73	#11764047	—	2	2	no	—
74	#11764047	—	2	2	no	—
75	#11764047	—	2	1	no	—
76	#11764047	—	2	1	no	—
77	#11764047	—	2	1	no	—
78	#11764047	—	2	1	no	—
79	C₂²×D₁₃₇	C₂²×C₄	1	2	yes	—
80	D₅₄₈	—	2	2	no	—
81	C₂²⋊D₁₃₇	—	2	4	no	—
82	C₂×Dic₁₃₇	—	2	2	no	—
83	C₂×Dic₁₃₇	—	2	1	no	—
84	C₂×Dic₁₃₇	—	2	1	no	—
85	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
88	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
89	C₂×Dic₁₃₇	C₂⁴	1	1	yes	characteristic
90	C₄×D₁₃₇	—	2	2	no	—
91	C₄×D₁₃₇	—	2	2	no	—
92	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
93	Q₈×C₁₃₇	—	2	2	no	—
94	C₂×C₄×C₁₃₇	—	2	2	no	—
95	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₁₃₇	—	2	1	no	—
100	C₂²×C₁₃₇	—	2	1	no	—
101	C₂²×C₁₃₇	#131	1	2	yes	—
102	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
103	C₂×D₁₃₇	—	2	2	no	—
104	C₂×D₁₃₇	—	2	2	no	—
105	C₂×D₁₃₇	—	2	2	no	—
106	C₄×C₁₃₇	—	2	2	no	—
107	C₄×C₁₃₇	—	2	1	no	—
108	C₄×C₁₃₇	—	2	2	no	—
109	C₄×C₁₃₇	#131	1	2	yes	—
110	Dic₁₃₇	—	2	1	no	—
111	Dic₁₃₇	—	2	1	no	—
112	Dic₁₃₇	—	2	1	no	—
113	Dic₁₃₇	—	2	2	no	—
114	Dic₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
115	Dic₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
116	C₂×C₁₃₇	—	2	2	no	—
117	C₂×C₁₃₇	#520	1	1	yes	characteristic
118	C₂×C₁₃₇	C₂×#131	1	1	yes	characteristic
119	C₂×C₁₃₇	C₂×#131	1	1	yes	characteristic
120	D₁₃₇	—	4	2	no	—
121	C₁₃₇	#2994	1	1	yes	characteristic, complemented, Sylow
122	#2994	—	137	1	no	maximal, Sylow
123	#529	—	137	1	no	—
124	#446	—	137	1	no	—
125	C₈×Q₈	—	137	1	no	—
126	#436	—	137	1	no	—
127	#435	—	137	2	no	—
128	C₈×D₄	—	137	2	no	—
129	C₈×D₄	—	137	1	no	—
130	#433	—	137	2	no	—
131	#431	—	137	1	no	—
132	#424	—	137	2	no	—
133	C₂×C₄⋊C₈	—	137	1	no	—
134	C₂×M₁₆	—	137	2	no	—
135	C₂²×C₈	—	137	2	no	—
136	C₂²×C₈	—	137	2	no	—
137	#126	—	137	2	no	—
138	#125	—	137	1	no	—
139	#124	—	137	1	no	—
140	#123	—	137	2	no	—
141	C₂²⋊Q₈	#10587739	1	1	yes	characteristic
142	#121	—	137	1	no	—
143	C₄×Q₈	—	137	1	no	—
144	C₄×D₄	—	137	1	no	—
145	C₄×D₄	—	137	2	no	—
146	#117	—	137	2	no	—
147	C₂×C₄²	—	137	1	no	—
148	C₄⋊C₈	—	137	2	no	—
149	C₄⋊C₈	—	137	1	no	—
150	C₄⋊C₈	—	137	1	no	—
151	C₄⋊C₈	—	137	2	no	—
152	C₄⋊C₈	—	137	1	no	—
153	C₄⋊C₈	—	137	2	no	—
154	C₄⋊C₈	—	137	1	no	—
155	C₂²⋊C₈	—	137	1	no	—
156	C₂²⋊C₈	—	137	1	no	—
157	C₂²⋊C₈	—	137	2	no	—
158	C₂²⋊C₈	—	137	2	no	—
159	#97	—	137	2	no	—
160	C₄×C₈	—	137	1	no	—
161	C₄×C₈	—	137	1	no	—
162	C₄×C₈	—	137	2	no	—
163	C₂×Q₈	C₂×#10587739	1	1	yes	characteristic
164	C₂×D₄	—	137	2	no	—
165	C₂×D₄	—	137	1	no	—
166	C₂²×C₄	—	137	1	no	—
167	C₂²×C₄	—	137	1	no	—
168	C₂²×C₄	—	137	2	no	—
169	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
170	M₁₆	—	548	2	no	—
171	C₂×C₈	—	548	2	no	—
172	C₂×C₈	—	548	2	no	—
173	C₂×C₈	—	137	1	no	—
174	C₂×C₈	—	137	1	no	—
175	C₂×C₈	—	137	1	no	—
176	C₂×C₈	—	137	2	no	—
177	C₂×C₈	—	137	1	no	—
178	C₂×C₈	—	137	2	no	—
179	C₄⋊C₄	—	137	2	no	—
180	C₄⋊C₄	—	137	2	no	—
181	C₄⋊C₄	—	137	1	no	—
182	C₄⋊C₄	—	137	2	no	—
183	C₄⋊C₄	C₂×#10587739	1	2	yes	—
184	C₄⋊C₄	C₂×#10587739	1	1	yes	characteristic
185	C₂²⋊C₄	—	137	2	no	—
186	C₂²⋊C₄	—	137	2	no	—
187	C₂²⋊C₄	—	137	2	no	—
188	C₂²⋊C₄	—	137	2	no	—
189	C₂²⋊C₄	C₂×#10587739	1	2	yes	—
190	C₄²	—	137	1	no	—
191	C₄²	—	137	1	no	—
192	C₄²	—	137	1	no	—
193	C₄²	—	137	2	no	—
194	C₄²	—	137	1	no	—
195	C₂³	—	137	2	no	—
196	C₂³	C₂²×#10587739	1	1	yes	characteristic
197	Q₈	—	2	2	no	—
198	D₄	—	274	4	no	—
199	D₄	—	274	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	274	2	no	—
202	C₂×C₄	—	274	2	no	—
203	C₂×C₄	—	274	2	no	—
204	C₂×C₄	—	274	1	no	—
205	C₂×C₄	—	274	1	no	—
206	C₂×C₄	—	137	1	no	—
207	C₂×C₄	—	137	1	no	—
208	C₂×C₄	—	137	1	no	—
209	C₂×C₄	—	137	2	no	—
210	C₂×C₄	—	137	1	no	—
211	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
212	C₂×C₄	C₂²×#10587739	1	2	yes	—
213	C₂×C₄	C₂²×#10587739	1	2	yes	—
214	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
215	C₈	—	274	2	no	—
216	C₈	—	274	1	no	—
217	C₈	—	274	2	no	—
218	C₈	—	274	1	no	—
219	C₈	—	274	1	no	—
220	C₈	—	274	1	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	274	2	no	—
224	C₂²	—	274	2	no	—
225	C₂²	—	274	2	no	—
226	C₂²	#17984176	1	2	yes	—
227	C₂²	C₂³×#10587739	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	2	no	—
231	C₄	—	274	1	no	—
232	C₄	—	274	1	no	—
233	C₄	—	274	1	no	—
234	C₄	—	274	2	no	—
235	C₄	—	137	1	no	—
236	C₄	—	137	1	no	—
237	C₄	#17984175	1	2	yes	—
238	C₂	—	2	2	no	—
239	C₂	—	548	2	no	—
240	C₂	#22578472	1	1	yes	characteristic
241	C₂	C₂×#17984175	1	1	yes	characteristic
242	C₂	C₂×#17984176	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	#17984062	C₄
cosocle	C₂×Dic₁₃₇	C₂⁴
hypercenter	C₂²⋊Q₈	#10587739
hypocenter	C₁₃₇	#2994
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. #22578454	C₂	semidirect
2. Q₈×#11764047	C₂	semidirect
3. #22578437	C₂	semidirect
4. #22578432	C₂	semidirect
5. #22578431	C₂	semidirect
6. #22578430	C₂	semidirect
7. #22578428	C₂	semidirect
8. #22578420	C₂	semidirect
9. #22578410	C₂	semidirect
10. C₂×C₄⋊#11764047	C₂	semidirect
121. C₁₃₇	#2994	semidirect

Maximal subgroups

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

subgroup	index
#22578454	2
Q₈×#11764047	2
#22578437	2
#22578432	2
#22578431	2
#22578430	2
#22578428	2
#22578420	2
#22578410	2
C₂×C₄⋊#11764047	2
#22577939	2
#2994	137