Subgroups — #34061642

Order 19072.

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	#23750676	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×#11924753	C₂	1	1	yes	maximal, characteristic, complemented
3	#23750659	C₂	1	1	yes	maximal, characteristic, complemented
4	#23750654	C₂	1	2	yes	maximal, complemented
5	#23750653	C₂	1	1	yes	maximal, characteristic, complemented
6	#23750652	C₂	1	2	yes	maximal, complemented
7	#23750650	C₂	1	2	yes	maximal, complemented
8	#23750642	C₂	1	1	yes	maximal, characteristic, complemented
9	#23750632	C₂	1	2	yes	maximal, complemented
10	C₂×C₄⋊#11924753	C₂	1	1	yes	maximal, characteristic, complemented
11	#23750161	C₂	1	1	yes	maximal, characteristic
12	C₂×#14558746	C₂²	1	2	yes	—
13	C₂×D₁₄₉⋊C₈	C₂²	1	2	yes	—
14	C₂²×#11924753	C₂²	1	2	yes	—
15	#18011978	C₂²	1	2	yes	—
16	#18011978	C₂²	1	1	yes	characteristic
17	#18011978	C₂²	1	1	yes	characteristic
18	C₄⋊#11924753	C₂²	1	1	yes	characteristic
19	C₄⋊#11924753	C₂²	1	1	yes	characteristic
20	C₄⋊#11924753	C₂²	1	2	yes	—
21	C₄⋊#11924753	C₂²	1	2	yes	—
22	#18011968	C₂²	1	2	yes	—
23	#18011968	C₂²	1	2	yes	—
24	C₂²⋊#11924753	C₂²	1	1	yes	characteristic
25	C₂²⋊#11924753	C₂²	1	1	yes	characteristic
26	#18011966	C₂²	1	2	yes	—
27	C₄×#11924753	C₂²	1	1	yes	characteristic
28	C₄×#11924753	C₂²	1	2	yes	—
29	C₄×#11924753	C₂²	1	1	yes	characteristic
30	#18011906	C₄	1	2	yes	—
31	#18011902	C₄	1	1	yes	characteristic
32	#18011900	C₄	1	1	yes	characteristic
33	#18011895	C₄	1	2	yes	—
34	#18011885	C₄	1	1	yes	characteristic
35	Q₈×Dic₁₄₉	C₂²	1	1	yes	characteristic
36	#18011873	C₂²	1	1	yes	characteristic
37	#18011871	C₂²	1	2	yes	—
38	#18011869	C₂²	1	2	yes	—
39	C₂×C₄×Dic₁₄₉	C₂²	1	1	yes	characteristic
40	C₂²⋊Q₈×C₁₄₉	C₄	1	1	yes	characteristic
41	#14558746	—	4	2	no	—
42	D₁₄₉⋊C₈	—	4	2	no	—
43	C₂×#11924753	—	4	2	no	—
44	C₂×#11924753	C₂³	1	1	yes	characteristic
45	C₂×#11924753	C₂³	1	1	yes	characteristic
46	C₂×#11924753	C₂³	1	1	yes	characteristic
47	C₂×#11924753	C₂³	1	1	yes	characteristic
48	C₂×#11924753	C₂³	1	2	yes	—
49	C₂×#11924753	C₂³	1	2	yes	—
50	C₂×D₅₉₆	C₂×C₄	1	1	yes	characteristic
51	C₂×C₂²⋊D₁₄₉	C₂×C₄	1	2	yes	—
52	C₂²×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
53	C₂²×Dic₁₄₉	C₂³	1	1	yes	characteristic
54	C₂×C₄×D₁₄₉	C₂³	1	2	yes	—
55	C₄⋊Dic₁₄₉	C₂×C₄	1	2	yes	—
56	C₄⋊Dic₁₄₉	C₂×C₄	1	2	yes	—
57	C₄⋊Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
58	#14558715	C₂×C₄	1	2	yes	—
59	#14558713	C₂×C₄	1	2	yes	—
60	#14558713	C₂×C₄	1	2	yes	—
61	#14558713	C₂×C₄	1	2	yes	—
62	#14558713	C₂×C₄	1	2	yes	—
63	C₄×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₁₄₉	C₂³	1	1	yes	characteristic
66	C₄×Dic₁₄₉	C₂³	1	2	yes	—
67	C₄×Dic₁₄₉	C₂³	1	1	yes	characteristic
68	C₂×Q₈×C₁₄₉	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
70	C₄⋊C₄×C₁₄₉	C₂×C₄	1	2	yes	—
71	C₄⋊C₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
72	C₂²⋊C₄×C₁₄₉	C₂×C₄	1	2	yes	—
73	#11924753	—	2	2	no	—
74	#11924753	—	2	2	no	—
75	#11924753	—	2	1	no	—
76	#11924753	—	2	1	no	—
77	#11924753	—	2	1	no	—
78	#11924753	—	2	1	no	—
79	C₂²×D₁₄₉	C₂²×C₄	1	2	yes	—
80	D₅₉₆	—	2	2	no	—
81	C₂²⋊D₁₄₉	—	2	4	no	—
82	C₂×Dic₁₄₉	—	2	2	no	—
83	C₂×Dic₁₄₉	—	2	1	no	—
84	C₂×Dic₁₄₉	—	2	1	no	—
85	C₂×Dic₁₄₉	C₂²×C₄	1	1	yes	characteristic
86	C₂×Dic₁₄₉	C₂²×C₄	1	1	yes	characteristic
87	C₂×Dic₁₄₉	C₂²×C₄	1	1	yes	characteristic
88	C₂×Dic₁₄₉	C₂²×C₄	1	2	yes	—
89	C₂×Dic₁₄₉	C₂⁴	1	1	yes	characteristic
90	C₄×D₁₄₉	—	2	2	no	—
91	C₄×D₁₄₉	—	2	2	no	—
92	C₂³×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
93	Q₈×C₁₄₉	—	2	2	no	—
94	C₂×C₄×C₁₄₉	—	2	2	no	—
95	C₂×C₄×C₁₄₉	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₁₄₉	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₁₄₉	—	2	1	no	—
100	C₂²×C₁₄₉	—	2	1	no	—
101	C₂²×C₁₄₉	#131	1	2	yes	—
102	C₂²×C₁₄₉	C₂³×C₄	1	1	yes	characteristic
103	C₂×D₁₄₉	—	2	2	no	—
104	C₂×D₁₄₉	—	2	2	no	—
105	C₂×D₁₄₉	—	2	2	no	—
106	C₄×C₁₄₉	—	2	2	no	—
107	C₄×C₁₄₉	—	2	1	no	—
108	C₄×C₁₄₉	—	2	2	no	—
109	C₄×C₁₄₉	#131	1	2	yes	—
110	Dic₁₄₉	—	2	1	no	—
111	Dic₁₄₉	—	2	1	no	—
112	Dic₁₄₉	—	2	1	no	—
113	Dic₁₄₉	—	2	2	no	—
114	Dic₁₄₉	H(5,𝔽₂)	1	1	yes	characteristic
115	Dic₁₄₉	2₋¹⁺⁴	1	1	yes	characteristic
116	C₂×C₁₄₉	—	2	2	no	—
117	C₂×C₁₄₉	#520	1	1	yes	characteristic
118	C₂×C₁₄₉	C₂×#131	1	1	yes	characteristic
119	C₂×C₁₄₉	C₂×#131	1	1	yes	characteristic
120	D₁₄₉	—	4	2	no	—
121	C₁₄₉	#2994	1	1	yes	characteristic, complemented, Sylow
122	#2994	—	149	1	no	maximal, Sylow
123	#529	—	149	1	no	—
124	#446	—	149	1	no	—
125	C₈×Q₈	—	149	1	no	—
126	#436	—	149	1	no	—
127	#435	—	149	2	no	—
128	C₈×D₄	—	149	2	no	—
129	C₈×D₄	—	149	1	no	—
130	#433	—	149	2	no	—
131	#431	—	149	1	no	—
132	#424	—	149	2	no	—
133	C₂×C₄⋊C₈	—	149	1	no	—
134	C₂×M₁₆	—	149	2	no	—
135	C₂²×C₈	—	149	2	no	—
136	C₂²×C₈	—	149	2	no	—
137	#126	—	149	2	no	—
138	#125	—	149	1	no	—
139	#124	—	149	1	no	—
140	#123	—	149	2	no	—
141	C₂²⋊Q₈	#10597010	1	1	yes	characteristic
142	#121	—	149	1	no	—
143	C₄×Q₈	—	149	1	no	—
144	C₄×D₄	—	149	1	no	—
145	C₄×D₄	—	149	2	no	—
146	#117	—	149	2	no	—
147	C₂×C₄²	—	149	1	no	—
148	C₄⋊C₈	—	149	2	no	—
149	C₄⋊C₈	—	149	1	no	—
150	C₄⋊C₈	—	149	1	no	—
151	C₄⋊C₈	—	149	2	no	—
152	C₄⋊C₈	—	149	1	no	—
153	C₄⋊C₈	—	149	2	no	—
154	C₄⋊C₈	—	149	1	no	—
155	C₂²⋊C₈	—	149	1	no	—
156	C₂²⋊C₈	—	149	1	no	—
157	C₂²⋊C₈	—	149	2	no	—
158	C₂²⋊C₈	—	149	2	no	—
159	#97	—	149	2	no	—
160	C₄×C₈	—	149	1	no	—
161	C₄×C₈	—	149	1	no	—
162	C₄×C₈	—	149	2	no	—
163	C₂×Q₈	C₂×#10597010	1	1	yes	characteristic
164	C₂×D₄	—	149	2	no	—
165	C₂×D₄	—	149	1	no	—
166	C₂²×C₄	—	149	1	no	—
167	C₂²×C₄	—	149	1	no	—
168	C₂²×C₄	—	149	2	no	—
169	C₂²×C₄	C₂×#10597010	1	1	yes	characteristic
170	M₁₆	—	596	2	no	—
171	C₂×C₈	—	596	2	no	—
172	C₂×C₈	—	596	2	no	—
173	C₂×C₈	—	149	1	no	—
174	C₂×C₈	—	149	1	no	—
175	C₂×C₈	—	149	1	no	—
176	C₂×C₈	—	149	2	no	—
177	C₂×C₈	—	149	1	no	—
178	C₂×C₈	—	149	2	no	—
179	C₄⋊C₄	—	149	2	no	—
180	C₄⋊C₄	—	149	2	no	—
181	C₄⋊C₄	—	149	1	no	—
182	C₄⋊C₄	—	149	2	no	—
183	C₄⋊C₄	C₂×#10597010	1	2	yes	—
184	C₄⋊C₄	C₂×#10597010	1	1	yes	characteristic
185	C₂²⋊C₄	—	149	2	no	—
186	C₂²⋊C₄	—	149	2	no	—
187	C₂²⋊C₄	—	149	2	no	—
188	C₂²⋊C₄	—	149	2	no	—
189	C₂²⋊C₄	C₂×#10597010	1	2	yes	—
190	C₄²	—	149	1	no	—
191	C₄²	—	149	2	no	—
192	C₄²	—	149	1	no	—
193	C₄²	—	149	1	no	—
194	C₄²	—	149	1	no	—
195	C₂³	—	149	2	no	—
196	C₂³	C₂²×#10597010	1	1	yes	characteristic
197	Q₈	—	2	2	no	—
198	D₄	—	298	4	no	—
199	D₄	—	298	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	298	2	no	—
202	C₂×C₄	—	298	2	no	—
203	C₂×C₄	—	298	2	no	—
204	C₂×C₄	—	298	1	no	—
205	C₂×C₄	—	298	1	no	—
206	C₂×C₄	—	149	1	no	—
207	C₂×C₄	—	149	1	no	—
208	C₂×C₄	—	149	1	no	—
209	C₂×C₄	—	149	2	no	—
210	C₂×C₄	—	149	1	no	—
211	C₂×C₄	C₂²×#10597010	1	1	yes	characteristic
212	C₂×C₄	C₂²×#10597010	1	2	yes	—
213	C₂×C₄	C₂²×#10597010	1	2	yes	—
214	C₂×C₄	C₂²×#10597010	1	1	yes	characteristic
215	C₈	—	298	2	no	—
216	C₈	—	298	1	no	—
217	C₈	—	298	2	no	—
218	C₈	—	298	1	no	—
219	C₈	—	298	1	no	—
220	C₈	—	298	1	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	298	2	no	—
224	C₂²	—	298	2	no	—
225	C₂²	—	298	2	no	—
226	C₂²	#18011999	1	2	yes	—
227	C₂²	C₂³×#10597010	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	2	no	—
231	C₄	—	298	1	no	—
232	C₄	—	298	1	no	—
233	C₄	—	298	1	no	—
234	C₄	—	298	2	no	—
235	C₄	—	149	1	no	—
236	C₄	—	149	1	no	—
237	C₄	#18011998	1	2	yes	—
238	C₂	—	2	2	no	—
239	C₂	—	596	2	no	—
240	C₂	#23750694	1	1	yes	characteristic
241	C₂	C₂×#18011998	1	1	yes	characteristic
242	C₂	C₂×#18011999	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#10597010
commutator, socle	C₂²×C₁₄₉	C₂³×C₄
Fitting	C₂²⋊Q₈×C₁₄₉	C₄
prime radical	#18011885	C₄
cosocle	C₂×Dic₁₄₉	C₂⁴
hypercenter	C₂²⋊Q₈	#10597010
hypocenter	C₁₄₉	#2994
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #23750676	C₂	semidirect
2. Q₈×#11924753	C₂	semidirect
3. #23750659	C₂	semidirect
4. #23750654	C₂	semidirect
5. #23750653	C₂	semidirect
6. #23750652	C₂	semidirect
7. #23750650	C₂	semidirect
8. #23750642	C₂	semidirect
9. #23750632	C₂	semidirect
10. C₂×C₄⋊#11924753	C₂	semidirect
121. C₁₄₉	#2994	semidirect

Maximal subgroups

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

subgroup	index
#23750676	2
Q₈×#11924753	2
#23750659	2
#23750654	2
#23750653	2
#23750652	2
#23750650	2
#23750642	2
#23750632	2
C₂×C₄⋊#11924753	2
#23750161	2
#2994	149