Subgroups — #36370514

Order 22144.

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	#26012542	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×#13063443	C₂	1	1	yes	maximal, characteristic, complemented
3	#26012525	C₂	1	1	yes	maximal, characteristic, complemented
4	#26012520	C₂	1	2	yes	maximal, complemented
5	#26012519	C₂	1	1	yes	maximal, characteristic, complemented
6	#26012518	C₂	1	2	yes	maximal, complemented
7	#26012516	C₂	1	2	yes	maximal, complemented
8	#26012508	C₂	1	1	yes	maximal, characteristic, complemented
9	#26012498	C₂	1	2	yes	maximal, complemented
10	C₂×C₄⋊#13063443	C₂	1	1	yes	maximal, characteristic, complemented
11	#26012027	C₂	1	1	yes	maximal, characteristic
12	C₂×#14583836	C₂²	1	2	yes	—
13	C₂×D₁₇₃⋊C₈	C₂²	1	2	yes	—
14	C₂²×#13063443	C₂²	1	2	yes	—
15	#19149746	C₂²	1	1	yes	characteristic
16	#19149746	C₂²	1	2	yes	—
17	#19149746	C₂²	1	1	yes	characteristic
18	C₄⋊#13063443	C₂²	1	2	yes	—
19	C₄⋊#13063443	C₂²	1	1	yes	characteristic
20	C₄⋊#13063443	C₂²	1	1	yes	characteristic
21	C₄⋊#13063443	C₂²	1	2	yes	—
22	#19149736	C₂²	1	2	yes	—
23	#19149736	C₂²	1	2	yes	—
24	C₂²⋊#13063443	C₂²	1	1	yes	characteristic
25	C₂²⋊#13063443	C₂²	1	1	yes	characteristic
26	#19149734	C₂²	1	2	yes	—
27	C₄×#13063443	C₂²	1	1	yes	characteristic
28	C₄×#13063443	C₂²	1	1	yes	characteristic
29	C₄×#13063443	C₂²	1	2	yes	—
30	#19149674	C₄	1	2	yes	—
31	#19149670	C₄	1	1	yes	characteristic
32	#19149668	C₄	1	1	yes	characteristic
33	#19149663	C₄	1	2	yes	—
34	#19149653	C₄	1	1	yes	characteristic
35	Q₈×Dic₁₇₃	C₂²	1	1	yes	characteristic
36	#19149641	C₂²	1	1	yes	characteristic
37	#19149639	C₂²	1	2	yes	—
38	#19149637	C₂²	1	2	yes	—
39	C₂×C₄×Dic₁₇₃	C₂²	1	1	yes	characteristic
40	C₂²⋊Q₈×C₁₇₃	C₄	1	1	yes	characteristic
41	#14583836	—	4	2	no	—
42	D₁₇₃⋊C₈	—	4	2	no	—
43	C₂×#13063443	—	4	2	no	—
44	C₂×#13063443	C₂³	1	1	yes	characteristic
45	C₂×#13063443	C₂³	1	1	yes	characteristic
46	C₂×#13063443	C₂³	1	1	yes	characteristic
47	C₂×#13063443	C₂³	1	1	yes	characteristic
48	C₂×#13063443	C₂³	1	2	yes	—
49	C₂×#13063443	C₂³	1	2	yes	—
50	C₂×D₆₉₂	C₂×C₄	1	1	yes	characteristic
51	C₂×C₂²⋊D₁₇₃	C₂×C₄	1	2	yes	—
52	C₂²×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
53	C₂²×Dic₁₇₃	C₂³	1	1	yes	characteristic
54	C₂×C₄×D₁₇₃	C₂³	1	2	yes	—
55	C₄⋊Dic₁₇₃	C₂×C₄	1	2	yes	—
56	C₄⋊Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
57	C₄⋊Dic₁₇₃	C₂×C₄	1	2	yes	—
58	#14583805	C₂×C₄	1	2	yes	—
59	#14583803	C₂×C₄	1	2	yes	—
60	#14583803	C₂×C₄	1	2	yes	—
61	#14583803	C₂×C₄	1	2	yes	—
62	#14583803	C₂×C₄	1	2	yes	—
63	C₄×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₁₇₃	C₂³	1	1	yes	characteristic
66	C₄×Dic₁₇₃	C₂³	1	1	yes	characteristic
67	C₄×Dic₁₇₃	C₂³	1	2	yes	—
68	C₂×Q₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
70	C₄⋊C₄×C₁₇₃	C₂×C₄	1	2	yes	—
71	C₄⋊C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
72	C₂²⋊C₄×C₁₇₃	C₂×C₄	1	2	yes	—
73	#13063443	—	2	2	no	—
74	#13063443	—	2	2	no	—
75	#13063443	—	2	1	no	—
76	#13063443	—	2	1	no	—
77	#13063443	—	2	1	no	—
78	#13063443	—	2	1	no	—
79	C₂²×D₁₇₃	C₂²×C₄	1	2	yes	—
80	D₆₉₂	—	2	2	no	—
81	C₂²⋊D₁₇₃	—	2	4	no	—
82	C₂×Dic₁₇₃	—	2	2	no	—
83	C₂×Dic₁₇₃	—	2	1	no	—
84	C₂×Dic₁₇₃	—	2	1	no	—
85	C₂×Dic₁₇₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×Dic₁₇₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×Dic₁₇₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×Dic₁₇₃	C₂²×C₄	1	2	yes	—
89	C₂×Dic₁₇₃	C₂⁴	1	1	yes	characteristic
90	C₄×D₁₇₃	—	2	2	no	—
91	C₄×D₁₇₃	—	2	2	no	—
92	C₂³×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
93	Q₈×C₁₇₃	—	2	2	no	—
94	C₂×C₄×C₁₇₃	—	2	2	no	—
95	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₁₇₃	—	2	1	no	—
100	C₂²×C₁₇₃	—	2	1	no	—
101	C₂²×C₁₇₃	#131	1	2	yes	—
102	C₂²×C₁₇₃	C₂³×C₄	1	1	yes	characteristic
103	C₂×D₁₇₃	—	2	2	no	—
104	C₂×D₁₇₃	—	2	2	no	—
105	C₂×D₁₇₃	—	2	2	no	—
106	C₄×C₁₇₃	—	2	2	no	—
107	C₄×C₁₇₃	—	2	1	no	—
108	C₄×C₁₇₃	—	2	2	no	—
109	C₄×C₁₇₃	#131	1	2	yes	—
110	Dic₁₇₃	—	2	1	no	—
111	Dic₁₇₃	—	2	1	no	—
112	Dic₁₇₃	—	2	1	no	—
113	Dic₁₇₃	—	2	2	no	—
114	Dic₁₇₃	H(5,𝔽₂)	1	1	yes	characteristic
115	Dic₁₇₃	2₋¹⁺⁴	1	1	yes	characteristic
116	C₂×C₁₇₃	—	2	2	no	—
117	C₂×C₁₇₃	#520	1	1	yes	characteristic
118	C₂×C₁₇₃	C₂×#131	1	1	yes	characteristic
119	C₂×C₁₇₃	C₂×#131	1	1	yes	characteristic
120	D₁₇₃	—	4	2	no	—
121	C₁₇₃	#2994	1	1	yes	characteristic, complemented, Sylow
122	#2994	—	173	1	no	maximal, Sylow
123	#529	—	173	1	no	—
124	#446	—	173	1	no	—
125	C₈×Q₈	—	173	1	no	—
126	#436	—	173	1	no	—
127	#435	—	173	2	no	—
128	C₈×D₄	—	173	2	no	—
129	C₈×D₄	—	173	1	no	—
130	#433	—	173	2	no	—
131	#431	—	173	1	no	—
132	#424	—	173	2	no	—
133	C₂×C₄⋊C₈	—	173	1	no	—
134	C₂×M₁₆	—	173	2	no	—
135	C₂²×C₈	—	173	2	no	—
136	C₂²×C₈	—	173	2	no	—
137	#126	—	173	2	no	—
138	#125	—	173	1	no	—
139	#124	—	173	1	no	—
140	#123	—	173	2	no	—
141	C₂²⋊Q₈	#10621924	1	1	yes	characteristic
142	#121	—	173	1	no	—
143	C₄×Q₈	—	173	1	no	—
144	C₄×D₄	—	173	1	no	—
145	C₄×D₄	—	173	2	no	—
146	#117	—	173	2	no	—
147	C₂×C₄²	—	173	1	no	—
148	C₄⋊C₈	—	173	2	no	—
149	C₄⋊C₈	—	173	1	no	—
150	C₄⋊C₈	—	173	1	no	—
151	C₄⋊C₈	—	173	1	no	—
152	C₄⋊C₈	—	173	2	no	—
153	C₄⋊C₈	—	173	2	no	—
154	C₄⋊C₈	—	173	1	no	—
155	C₂²⋊C₈	—	173	1	no	—
156	C₂²⋊C₈	—	173	1	no	—
157	C₂²⋊C₈	—	173	2	no	—
158	C₂²⋊C₈	—	173	2	no	—
159	#97	—	173	2	no	—
160	C₄×C₈	—	173	1	no	—
161	C₄×C₈	—	173	1	no	—
162	C₄×C₈	—	173	2	no	—
163	C₂×Q₈	C₂×#10621924	1	1	yes	characteristic
164	C₂×D₄	—	173	2	no	—
165	C₂×D₄	—	173	1	no	—
166	C₂²×C₄	—	173	1	no	—
167	C₂²×C₄	—	173	1	no	—
168	C₂²×C₄	—	173	2	no	—
169	C₂²×C₄	C₂×#10621924	1	1	yes	characteristic
170	M₁₆	—	692	2	no	—
171	C₂×C₈	—	173	1	no	—
172	C₂×C₈	—	173	1	no	—
173	C₂×C₈	—	173	1	no	—
174	C₂×C₈	—	173	2	no	—
175	C₂×C₈	—	173	1	no	—
176	C₂×C₈	—	173	2	no	—
177	C₂×C₈	—	692	2	no	—
178	C₂×C₈	—	692	2	no	—
179	C₄⋊C₄	—	173	2	no	—
180	C₄⋊C₄	—	173	2	no	—
181	C₄⋊C₄	—	173	1	no	—
182	C₄⋊C₄	—	173	2	no	—
183	C₄⋊C₄	C₂×#10621924	1	2	yes	—
184	C₄⋊C₄	C₂×#10621924	1	1	yes	characteristic
185	C₂²⋊C₄	—	173	2	no	—
186	C₂²⋊C₄	—	173	2	no	—
187	C₂²⋊C₄	—	173	2	no	—
188	C₂²⋊C₄	—	173	2	no	—
189	C₂²⋊C₄	C₂×#10621924	1	2	yes	—
190	C₄²	—	173	1	no	—
191	C₄²	—	173	2	no	—
192	C₄²	—	173	1	no	—
193	C₄²	—	173	1	no	—
194	C₄²	—	173	1	no	—
195	C₂³	—	173	2	no	—
196	C₂³	C₂²×#10621924	1	1	yes	characteristic
197	Q₈	—	2	2	no	—
198	D₄	—	346	4	no	—
199	D₄	—	346	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	346	2	no	—
202	C₂×C₄	—	346	2	no	—
203	C₂×C₄	—	346	2	no	—
204	C₂×C₄	—	346	1	no	—
205	C₂×C₄	—	346	1	no	—
206	C₂×C₄	—	173	1	no	—
207	C₂×C₄	—	173	1	no	—
208	C₂×C₄	—	173	1	no	—
209	C₂×C₄	—	173	2	no	—
210	C₂×C₄	—	173	1	no	—
211	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
212	C₂×C₄	C₂²×#10621924	1	2	yes	—
213	C₂×C₄	C₂²×#10621924	1	2	yes	—
214	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
215	C₈	—	346	1	no	—
216	C₈	—	346	1	no	—
217	C₈	—	346	2	no	—
218	C₈	—	346	1	no	—
219	C₈	—	346	2	no	—
220	C₈	—	346	1	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	346	2	no	—
224	C₂²	—	346	2	no	—
225	C₂²	—	346	2	no	—
226	C₂²	#19149767	1	2	yes	—
227	C₂²	C₂³×#10621924	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	2	no	—
231	C₄	—	346	1	no	—
232	C₄	—	346	1	no	—
233	C₄	—	346	1	no	—
234	C₄	—	346	2	no	—
235	C₄	—	173	1	no	—
236	C₄	—	173	1	no	—
237	C₄	#19149766	1	2	yes	—
238	C₂	—	2	2	no	—
239	C₂	—	692	2	no	—
240	C₂	#26012560	1	1	yes	characteristic
241	C₂	C₂×#19149766	1	1	yes	characteristic
242	C₂	C₂×#19149767	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#10621924
commutator, socle	C₂²×C₁₇₃	C₂³×C₄
Fitting	C₂²⋊Q₈×C₁₇₃	C₄
prime radical	#19149653	C₄
cosocle	C₂×Dic₁₇₃	C₂⁴
hypercenter	C₂²⋊Q₈	#10621924
hypocenter	C₁₇₃	#2994
LCS	G ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #26012542	C₂	semidirect
2. Q₈×#13063443	C₂	semidirect
3. #26012525	C₂	semidirect
4. #26012520	C₂	semidirect
5. #26012519	C₂	semidirect
6. #26012518	C₂	semidirect
7. #26012516	C₂	semidirect
8. #26012508	C₂	semidirect
9. #26012498	C₂	semidirect
10. C₂×C₄⋊#13063443	C₂	semidirect
121. C₁₇₃	#2994	semidirect

Maximal subgroups

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

subgroup	index
#26012542	2
Q₈×#13063443	2
#26012525	2
#26012520	2
#26012519	2
#26012518	2
#26012516	2
#26012508	2
#26012498	2
C₂×C₄⋊#13063443	2
#26012027	2
#2994	173