Subgroups — #34061516

Order 19072.

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	#23750677	C₂	1	2	yes	maximal, complemented
2	#23750656	C₂	1	2	yes	maximal, complemented
3	#23750656	C₂	1	2	yes	maximal, complemented
4	D₄×#11924753	C₂	1	2	yes	maximal, complemented
5	#23750649	C₂	1	2	yes	maximal, complemented
6	#23750642	C₂	1	1	yes	maximal, characteristic, complemented
7	#23750589	C₂	1	2	yes	maximal, complemented
8	C₂×#18011966	C₂	1	1	yes	maximal, characteristic, complemented
9	#23750156	C₂	1	1	yes	maximal, characteristic
10	C₂×#14558745	C₂²	1	2	yes	—
11	C₂²×#11924753	C₂²	1	2	yes	—
12	C₂²×#11924753	C₂²	1	2	yes	—
13	#18011978	C₂²	1	2	yes	—
14	#18011978	C₂²	1	2	yes	—
15	C₄⋊#11924753	C₂²	1	2	yes	—
16	C₂²⋊#11924753	C₂²	1	1	yes	characteristic
17	C₂²⋊#11924753	C₂²	1	2	yes	—
18	C₂²⋊#11924753	C₂²	1	2	yes	—
19	C₂²⋊#11924753	C₂²	1	1	yes	characteristic
20	#18011966	C₂²	1	2	yes	—
21	#18011966	C₂²	1	2	yes	—
22	#18011966	C₂²	1	1	yes	characteristic
23	#18011966	C₂²	1	1	yes	characteristic
24	C₄×#11924753	C₂²	1	2	yes	—
25	C₄×#11924753	C₂²	1	2	yes	—
26	#18011904	C₄	1	2	yes	—
27	#18011902	C₄	1	1	yes	characteristic
28	#18011898	C₄	1	1	yes	characteristic
29	#18011892	C₄	1	2	yes	—
30	#18011890	C₄	1	1	yes	characteristic
31	#18011876	C₂²	1	1	yes	characteristic
32	D₄×Dic₁₄₉	C₂²	1	1	yes	characteristic
33	D₄×Dic₁₄₉	C₂²	1	2	yes	—
34	#18011867	C₂²	1	2	yes	—
35	C₂×C₄×Dic₁₄₉	C₂²	1	1	yes	characteristic
36	#121×C₁₄₉	C₄	1	1	yes	characteristic
37	#14558745	—	2	4	no	—
38	C₂×#11924753	—	2	4	no	—
39	C₂×#11924753	—	2	4	no	—
40	C₂×#11924753	C₂³	1	2	yes	—
41	C₂×#11924753	C₂³	1	2	yes	—
42	C₂×#11924753	C₂³	1	2	yes	—
43	C₂×#11924753	C₂³	1	2	yes	—
44	C₂×Dic₂₉₈	C₂×C₄	1	1	yes	characteristic
45	C₂²×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
46	C₂²×Dic₁₄₉	C₂³	1	1	yes	characteristic
47	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
48	C₄⋊Dic₁₄₉	C₂×C₄	1	2	yes	—
49	C₄⋊Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
50	#14558715	C₂×C₄	1	2	yes	—
51	#14558715	C₂×C₄	1	2	yes	—
52	#14558715	C₂×C₄	1	2	yes	—
53	C₂²⋊Dic₁₄₉	C₂×C₄	1	2	yes	—
54	C₂²⋊Dic₁₄₉	C₂×C₄	1	2	yes	—
55	C₂²⋊Dic₁₄₉	C₂×C₄	1	2	yes	—
56	C₂²⋊Dic₁₄₉	C₂×C₄	1	2	yes	—
57	C₄×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
58	C₄×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
59	C₄×Dic₁₄₉	C₂³	1	1	yes	characteristic
60	C₄×Dic₁₄₉	C₂³	1	2	yes	—
61	C₄×Dic₁₄₉	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₁₄₉	C₂×C₄	1	2	yes	—
63	C₂×D₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
65	C₄⋊C₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
66	C₂²⋊C₄×C₁₄₉	C₂×C₄	1	2	yes	—
67	#11924753	—	2	2	no	—
68	#11924753	—	2	2	no	—
69	#11924753	—	2	2	no	—
70	#11924753	#41	1	4	yes	—
71	Dic₂₉₈	—	4	1	no	—
72	C₂×Dic₁₄₉	—	2	2	no	—
73	C₂×Dic₁₄₉	—	2	2	no	—
74	C₂×Dic₁₄₉	—	2	2	no	—
75	C₂×Dic₁₄₉	—	2	2	no	—
76	C₂×Dic₁₄₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×Dic₁₄₉	C₂²×C₄	1	1	yes	characteristic
78	C₂×Dic₁₄₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×Dic₁₄₉	C₂²×C₄	1	2	yes	—
80	C₂×Dic₁₄₉	C₂²×C₄	1	2	yes	—
81	C₂×Dic₁₄₉	C₂⁴	1	1	yes	characteristic
82	C₂³×C₁₄₉	C₂²×C₄	1	2	yes	—
83	C₂³×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
84	D₄×C₁₄₉	—	2	4	no	—
85	D₄×C₁₄₉	—	4	1	no	—
86	C₂×C₄×C₁₄₉	—	4	1	no	—
87	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₄₉	C₂²×C₄	1	2	yes	—
90	C₂²×C₁₄₉	—	2	1	no	—
91	C₂²×C₁₄₉	—	2	1	no	—
92	C₂²×C₁₄₉	—	2	2	no	—
93	C₂²×C₁₄₉	—	2	2	no	—
94	C₂²×C₁₄₉	—	2	2	no	—
95	C₂²×C₁₄₉	#131	1	2	yes	—
96	C₂²×C₁₄₉	C₂³×C₄	1	1	yes	characteristic
97	C₄×C₁₄₉	—	2	2	no	—
98	C₄×C₁₄₉	—	2	1	no	—
99	C₄×C₁₄₉	—	2	1	no	—
100	Dic₁₄₉	—	2	1	no	—
101	Dic₁₄₉	—	2	2	no	—
102	Dic₁₄₉	—	2	1	no	—
103	Dic₁₄₉	—	2	1	no	—
104	Dic₁₄₉	—	2	2	no	—
105	Dic₁₄₉	C₂×#41	1	2	yes	—
106	C₂×C₁₄₉	—	2	2	no	—
107	C₂×C₁₄₉	—	4	2	no	—
108	C₂×C₁₄₉	C₄×#41	1	1	yes	characteristic
109	C₂×C₁₄₉	C₂×#131	1	1	yes	characteristic
110	C₂×C₁₄₉	#568	1	1	yes	characteristic
111	C₁₄₉	#2968	1	1	yes	characteristic, complemented, Sylow
112	#2968	—	149	1	no	maximal, Sylow
113	#529	—	149	1	no	—
114	#446	—	149	2	no	—
115	#435	—	149	2	no	—
116	#435	—	149	2	no	—
117	C₈×D₄	—	149	2	no	—
118	#433	—	149	2	no	—
119	#431	—	149	1	no	—
120	#405	—	149	2	no	—
121	C₂×#97	—	149	1	no	—
122	C₂×M₁₆	—	149	2	no	—
123	C₂²×C₈	—	149	2	no	—
124	C₂²×C₈	—	149	2	no	—
125	#126	—	149	2	no	—
126	#125	—	149	1	no	—
127	#124	—	149	1	no	—
128	#123	—	149	2	no	—
129	C₂²⋊Q₈	—	149	1	no	—
130	#121	#10597010	1	1	yes	characteristic
131	C₄×Q₈	—	149	1	no	—
132	C₄×D₄	—	149	2	no	—
133	C₄×D₄	—	149	1	no	—
134	#117	—	149	2	no	—
135	C₂×C₄²	—	149	1	no	—
136	C₄⋊C₈	—	149	2	no	—
137	C₄⋊C₈	—	149	2	no	—
138	C₄⋊C₈	—	149	2	no	—
139	C₂²⋊C₈	—	149	1	no	—
140	C₂²⋊C₈	—	149	1	no	—
141	C₂²⋊C₈	—	149	2	no	—
142	C₂²⋊C₈	—	149	2	no	—
143	#97	—	149	1	no	—
144	#97	—	149	1	no	—
145	#97	—	149	2	no	—
146	#97	—	149	2	no	—
147	C₄×C₈	—	149	2	no	—
148	C₄×C₈	—	149	2	no	—
149	C₂×Q₈	—	149	1	no	—
150	C₂×D₄	C₂×#10597010	1	2	yes	—
151	C₂×D₄	C₂×#10597010	1	1	yes	characteristic
152	C₂²×C₄	—	149	1	no	—
153	C₂²×C₄	—	149	1	no	—
154	C₂²×C₄	—	149	2	no	—
155	C₂²×C₄	C₂×#10597010	1	1	yes	characteristic
156	M₁₆	—	298	4	no	—
157	C₂×C₈	—	298	4	no	—
158	C₂×C₈	—	298	4	no	—
159	C₂×C₈	—	149	2	no	—
160	C₂×C₈	—	149	2	no	—
161	C₂×C₈	—	149	2	no	—
162	C₂×C₈	—	149	2	no	—
163	C₄⋊C₄	—	149	2	no	—
164	C₄⋊C₄	—	149	2	no	—
165	C₄⋊C₄	—	149	1	no	—
166	C₄⋊C₄	—	149	2	no	—
167	C₄⋊C₄	—	149	2	no	—
168	C₄⋊C₄	C₂×#10597010	1	1	yes	characteristic
169	C₂²⋊C₄	—	149	2	no	—
170	C₂²⋊C₄	—	149	2	no	—
171	C₂²⋊C₄	—	149	2	no	—
172	C₂²⋊C₄	—	149	2	no	—
173	C₂²⋊C₄	C₂×#10597010	1	2	yes	—
174	C₄²	—	149	1	no	—
175	C₄²	—	149	2	no	—
176	C₄²	—	149	1	no	—
177	C₄²	—	149	1	no	—
178	C₄²	—	149	1	no	—
179	C₂³	C₂²×#10597010	1	2	yes	—
180	C₂³	C₂²×#10597010	1	1	yes	characteristic
181	Q₈	—	596	1	no	—
182	D₄	—	2	4	no	—
183	D₄	—	4	1	no	—
184	C₂×C₄	—	4	1	no	—
185	C₂×C₄	—	298	2	no	—
186	C₂×C₄	—	298	2	no	—
187	C₂×C₄	—	298	2	no	—
188	C₂×C₄	—	298	2	no	—
189	C₂×C₄	—	149	1	no	—
190	C₂×C₄	—	149	1	no	—
191	C₂×C₄	—	149	1	no	—
192	C₂×C₄	—	149	2	no	—
193	C₂×C₄	—	149	2	no	—
194	C₂×C₄	—	149	1	no	—
195	C₂×C₄	C₂²×#10597010	1	1	yes	characteristic
196	C₂×C₄	C₂²×#10597010	1	1	yes	characteristic
197	C₂×C₄	C₂²×#10597010	1	2	yes	—
198	C₈	—	298	2	no	—
199	C₈	—	298	2	no	—
200	C₈	—	298	2	no	—
201	C₈	—	149	4	no	—
202	C₂²	—	2	1	no	—
203	C₂²	—	2	1	no	—
204	C₂²	—	2	2	no	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	#18011998	1	2	yes	—
208	C₂²	C₂³×#10597010	1	1	yes	characteristic
209	C₄	—	2	2	no	—
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	298	1	no	—
213	C₄	—	298	2	no	—
214	C₄	—	298	1	no	—
215	C₄	—	298	1	no	—
216	C₄	—	298	2	no	—
217	C₄	—	149	2	no	—
218	C₂	—	2	2	no	—
219	C₂	—	4	2	no	—
220	C₂	#41×#10597010	1	1	yes	characteristic
221	C₂	C₂×#18011998	1	1	yes	characteristic
222	C₂	#23750703	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#10597010
commutator, socle	C₂²×C₁₄₉	C₂³×C₄
Fitting, prime radical	#121×C₁₄₉	C₄
cosocle	C₂×Dic₁₄₉	C₂⁴
hypercenter	#121	#10597010
hypocenter	C₁₄₉	#2968
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ #121

Complements

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

N	H ≅ G/N	type
1. #23750677	C₂	semidirect
2. #23750656	C₂	semidirect
3. #23750656	C₂	semidirect
4. D₄×#11924753	C₂	semidirect
5. #23750649	C₂	semidirect
6. #23750642	C₂	semidirect
7. #23750589	C₂	semidirect
8. C₂×#18011966	C₂	semidirect
111. C₁₄₉	#2968	semidirect

Maximal subgroups

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

subgroup	index
#23750677	2
#23750656	2
D₄×#11924753	2
#23750649	2
#23750642	2
#23750589	2
C₂×#18011966	2
#23750156	2
#2968	149