Subgroups — #31797584

Order 17536.

356 conjugacy classes of subgroups.

The subgroup lattice has 234 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	#22578454	C₂	1	1	yes	maximal, characteristic, complemented
3	#22578453	C₂	1	1	yes	maximal, characteristic, complemented
4	#22578453	C₂	1	2	yes	maximal, complemented
5	#22578452	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₈×#11764047	C₂	1	1	yes	maximal, characteristic, complemented
7	#22578409	C₂	1	2	yes	maximal
8	C₂×C₄⋊#11764047	C₂	1	1	yes	maximal, characteristic
9	#22578367	C₂	1	2	yes	maximal
10	C₄×#14555519	C₂	1	1	yes	maximal, characteristic
11	#22577955	C₂	1	1	yes	maximal, characteristic
12	C₂×#14555519	C₂²	1	1	yes	characteristic
13	C₂×#14555519	C₂²	1	1	yes	characteristic
14	C₂²×#11764047	C₂²	1	2	yes	—
15	#17984155	C₂²	1	2	yes	—
16	#17984155	C₂²	1	2	yes	—
17	C₄⋊#11764047	C₂²	1	1	yes	characteristic
18	C₄⋊#11764047	C₂²	1	2	yes	—
19	C₄⋊#11764047	C₂²	1	2	yes	—
20	C₄⋊#11764047	C₂²	1	1	yes	characteristic
21	C₄⋊#11764047	C₂²	1	2	yes	—
22	#17984143	C₂²	1	1	yes	characteristic
23	#17984143	C₂²	1	2	yes	—
24	#17984143	C₂²	1	2	yes	—
25	#17984143	C₂²	1	1	yes	characteristic
26	C₄×#11764047	C₂²	1	1	yes	characteristic
27	C₄×#11764047	C₂²	1	2	yes	—
28	C₄×#11764047	C₂²	1	2	yes	—
29	C₄×#11764047	C₂²	1	1	yes	characteristic
30	#17984087	C₄	1	1	yes	characteristic
31	#17984086	C₄	1	1	yes	characteristic
32	#17984078	C₄	1	2	yes	—
33	#17984067	C₄	1	1	yes	characteristic
34	#17984065	C₄	1	2	yes	—
35	Q₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
36	#17984053	C₂²	1	1	yes	characteristic
37	#17984053	C₂²	1	2	yes	—
38	#17984044	C₂²	1	2	yes	—
39	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
40	C₂²⋊Q₈×C₁₃₇	C₄	1	1	yes	characteristic
41	#14555519	—	2	2	no	—
42	#14555519	Q₈	1	4	yes	complemented
43	C₂×#11764047	—	2	4	no	—
44	C₂×#11764047	C₂³	1	2	yes	—
45	C₂×#11764047	C₂³	1	1	yes	characteristic
46	C₂×#11764047	C₂³	1	1	yes	characteristic
47	C₂×#11764047	C₂³	1	2	yes	—
48	C₂×#11764047	C₂³	1	1	yes	characteristic
49	C₂×#11764047	C₂³	1	1	yes	characteristic
50	C₂×Dic₂₇₄	C₂×C₄	1	1	yes	characteristic
51	C₂×Dic₂₇₄	C₂×C₄	1	2	yes	—
52	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
53	C₂²×Dic₁₃₇	C₂³	1	1	yes	characteristic
54	C₄⋊Dic₁₃₇	C₂×C₄	1	2	yes	—
55	C₄⋊Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
56	#14555489	C₂×C₄	1	2	yes	—
57	#14555489	C₂×C₄	1	2	yes	—
58	#14555489	C₂×C₄	1	2	yes	—
59	#14555489	C₂×C₄	1	2	yes	—
60	#14555489	C₂×C₄	1	2	yes	—
61	C₂²⋊Dic₁₃₇	C₂×C₄	1	2	yes	—
62	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
63	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
65	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
66	C₄×Dic₁₃₇	C₂³	1	2	yes	—
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	C₂×Q₈	1	2	yes	—
78	#11764047	C₂×Q₈	1	2	yes	—
79	Dic₂₇₄	—	2	2	no	—
80	Dic₂₇₄	—	4	2	no	—
81	C₂×Dic₁₃₇	—	4	1	no	—
82	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
85	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
86	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×Dic₁₃₇	C₂×Q₈	1	2	yes	—
88	C₂×Dic₁₃₇	#41	1	2	yes	—
89	C₂×Dic₁₃₇	C₂⁴	1	1	yes	characteristic
90	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
91	Q₈×C₁₃₇	—	2	2	no	—
92	C₂×C₄×C₁₃₇	—	2	2	no	—
93	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
95	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₁₃₇	—	2	1	no	—
98	C₂²×C₁₃₇	—	2	1	no	—
99	C₂²×C₁₃₇	C₄×Q₈	1	2	yes	—
100	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
101	C₄×C₁₃₇	—	2	1	no	—
102	C₄×C₁₃₇	—	2	2	no	—
103	C₄×C₁₃₇	—	2	2	no	—
104	C₄×C₁₃₇	#131	1	2	yes	—
105	Dic₁₃₇	—	2	2	no	—
106	Dic₁₃₇	—	2	1	no	—
107	Dic₁₃₇	—	2	1	no	—
108	Dic₁₃₇	—	2	2	no	—
109	Dic₁₃₇	C₄×Q₈	1	2	yes	—
110	Dic₁₃₇	C₂²×Q₈	1	1	yes	characteristic
111	Dic₁₃₇	C₂×#41	1	1	yes	characteristic
112	C₂×C₁₃₇	—	2	2	no	—
113	C₂×C₁₃₇	C₂×C₄×Q₈	1	1	yes	characteristic
114	C₂×C₁₃₇	C₂×#131	1	1	yes	characteristic
115	C₂×C₁₃₇	#568	1	1	yes	characteristic
116	C₁₃₇	#2962	1	1	yes	characteristic, complemented, Sylow
117	#2962	—	137	1	no	maximal, Sylow
118	#533	—	137	1	no	—
119	#446	—	137	1	no	—
120	#446	—	137	2	no	—
121	#446	—	137	1	no	—
122	#446	—	137	2	no	—
123	C₈×Q₈	—	137	1	no	—
124	C₈×Q₈	—	137	1	no	—
125	#424	—	137	2	no	—
126	C₂×C₄⋊C₈	—	137	1	no	—
127	#405	—	137	2	no	—
128	C₄×M₁₆	—	137	1	no	—
129	C₂×M₁₆	—	137	1	no	—
130	C₂×M₁₆	—	137	1	no	—
131	C₂²×C₈	—	137	2	no	—
132	C₄⋊Q₈	—	137	1	no	—
133	C₄⋊Q₈	—	137	1	no	—
134	#125	—	137	2	no	—
135	C₂²⋊Q₈	—	137	1	no	—
136	C₂²⋊Q₈	—	137	2	no	—
137	C₂²⋊Q₈	#10587739	1	1	yes	characteristic
138	C₄×Q₈	—	137	1	no	—
139	C₄×Q₈	—	137	1	no	—
140	C₄×Q₈	—	137	2	no	—
141	#117	—	137	2	no	—
142	C₂×C₄²	—	137	1	no	—
143	C₄⋊C₈	—	137	1	no	—
144	C₄⋊C₈	—	137	1	no	—
145	C₄⋊C₈	—	137	2	no	—
146	C₄⋊C₈	—	137	2	no	—
147	C₄⋊C₈	—	137	2	no	—
148	C₄⋊C₈	—	137	2	no	—
149	C₄⋊C₈	—	137	2	no	—
150	#97	—	137	1	no	—
151	#97	—	137	1	no	—
152	#97	—	137	2	no	—
153	#97	—	137	2	no	—
154	C₄×C₈	—	137	1	no	—
155	C₄×C₈	—	137	1	no	—
156	C₄×C₈	—	137	2	no	—
157	C₄×C₈	—	137	2	no	—
158	C₂×Q₈	—	137	1	no	—
159	C₂×Q₈	—	137	2	no	—
160	C₂×Q₈	C₂×#10587739	1	1	yes	characteristic
161	C₂²×C₄	—	137	1	no	—
162	C₂²×C₄	—	137	1	no	—
163	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
164	M₁₆	—	274	2	no	—
165	M₁₆	—	137	4	no	—
166	C₂×C₈	—	274	4	no	—
167	C₂×C₈	—	137	2	no	—
168	C₂×C₈	—	137	1	no	—
169	C₂×C₈	—	137	1	no	—
170	C₂×C₈	—	137	2	no	—
171	C₂×C₈	—	137	1	no	—
172	C₂×C₈	—	137	1	no	—
173	C₄⋊C₄	—	137	2	no	—
174	C₄⋊C₄	—	137	2	no	—
175	C₄⋊C₄	—	137	2	no	—
176	C₄⋊C₄	—	137	2	no	—
177	C₄⋊C₄	—	137	2	no	—
178	C₄⋊C₄	—	137	2	no	—
179	C₄⋊C₄	—	137	1	no	—
180	C₄⋊C₄	C₂×#10587739	1	2	yes	—
181	C₄⋊C₄	C₂×#10587739	1	1	yes	characteristic
182	C₂²⋊C₄	—	137	2	no	—
183	C₂²⋊C₄	C₂×#10587739	1	2	yes	—
184	C₄²	—	137	1	no	—
185	C₄²	—	137	1	no	—
186	C₄²	—	137	1	no	—
187	C₄²	—	137	2	no	—
188	C₄²	—	137	2	no	—
189	C₄²	—	137	1	no	—
190	C₂³	C₂²×#10587739	1	1	yes	characteristic
191	Q₈	—	2	2	no	—
192	Q₈	—	274	2	no	—
193	Q₈	—	548	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	548	1	no	—
196	C₂×C₄	—	137	1	no	—
197	C₂×C₄	—	137	1	no	—
198	C₂×C₄	—	137	2	no	—
199	C₂×C₄	—	137	2	no	—
200	C₂×C₄	—	137	2	no	—
201	C₂×C₄	—	137	2	no	—
202	C₂×C₄	—	137	1	no	—
203	C₂×C₄	—	137	1	no	—
204	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
205	C₂×C₄	C₂²×#10587739	1	2	yes	—
206	C₂×C₄	C₂²×#10587739	1	2	yes	—
207	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
208	C₈	—	274	2	no	—
209	C₈	—	274	2	no	—
210	C₈	—	274	1	no	—
211	C₈	—	274	1	no	—
212	C₈	—	137	2	no	—
213	C₈	—	137	2	no	—
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	Q₈×#10587739	1	2	yes	—
217	C₂²	C₂³×#10587739	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	2	2	no	—
220	C₄	—	2	2	no	—
221	C₄	—	274	2	no	—
222	C₄	—	274	1	no	—
223	C₄	—	274	1	no	—
224	C₄	—	274	2	no	—
225	C₄	—	137	1	no	—
226	C₄	—	137	1	no	—
227	C₄	—	137	2	no	—
228	C₄	#17984175	1	2	yes	—
229	C₂	—	2	2	no	—
230	C₂	C₂×Q₈×#10587739	1	1	yes	characteristic
231	C₂	C₂×#17984175	1	1	yes	characteristic
232	C₂	#22578482	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. #22578454	C₂	semidirect
3. #22578453	C₂	semidirect
4. #22578453	C₂	semidirect
5. #22578452	C₂	semidirect
6. Q₈×#11764047	C₂	semidirect
42. #14555519	Q₈	semidirect
116. 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
#22578453	2
#22578452	2
Q₈×#11764047	2
#22578409	2
C₂×C₄⋊#11764047	2
#22578367	2
C₄×#14555519	2
#22577955	2
#2962	137