Subgroups — #27158687

Order 12416.

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	#20261786	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×#11715897	C₂	1	1	yes	maximal, characteristic, complemented
3	#20261769	C₂	1	1	yes	maximal, characteristic, complemented
4	#20261764	C₂	1	2	yes	maximal, complemented
5	#20261763	C₂	1	1	yes	maximal, characteristic, complemented
6	#20261762	C₂	1	2	yes	maximal, complemented
7	#20261760	C₂	1	2	yes	maximal, complemented
8	#20261752	C₂	1	1	yes	maximal, characteristic, complemented
9	#20261742	C₂	1	2	yes	maximal, complemented
10	C₂×C₄⋊#11715897	C₂	1	1	yes	maximal, characteristic, complemented
11	#20261271	C₂	1	1	yes	maximal, characteristic
12	C₂×#13095521	C₂²	1	2	yes	—
13	C₂×#13095519	C₂²	1	2	yes	—
14	C₂²×#11715897	C₂²	1	2	yes	—
15	#15690877	C₂²	1	1	yes	characteristic
16	#15690877	C₂²	1	2	yes	—
17	#15690877	C₂²	1	1	yes	characteristic
18	C₄⋊#11715897	C₂²	1	2	yes	—
19	C₄⋊#11715897	C₂²	1	1	yes	characteristic
20	C₄⋊#11715897	C₂²	1	1	yes	characteristic
21	C₄⋊#11715897	C₂²	1	2	yes	—
22	#15690867	C₂²	1	2	yes	—
23	#15690867	C₂²	1	2	yes	—
24	C₂²⋊#11715897	C₂²	1	1	yes	characteristic
25	C₂²⋊#11715897	C₂²	1	1	yes	characteristic
26	#15690865	C₂²	1	2	yes	—
27	C₄×#11715897	C₂²	1	1	yes	characteristic
28	C₄×#11715897	C₂²	1	1	yes	characteristic
29	C₄×#11715897	C₂²	1	2	yes	—
30	#15690805	C₄	1	2	yes	—
31	#15690801	C₄	1	1	yes	characteristic
32	#15690799	C₄	1	1	yes	characteristic
33	#15690794	C₄	1	2	yes	—
34	#15690784	C₄	1	1	yes	characteristic
35	Q₈×Dic₉₇	C₂²	1	1	yes	characteristic
36	#15690772	C₂²	1	1	yes	characteristic
37	#15690770	C₂²	1	2	yes	—
38	#15690768	C₂²	1	2	yes	—
39	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
40	C₂²⋊Q₈×C₉₇	C₄	1	1	yes	characteristic
41	#13095521	—	4	2	no	—
42	#13095519	—	4	2	no	—
43	C₂×#11715897	—	4	2	no	—
44	C₂×#11715897	C₂³	1	1	yes	characteristic
45	C₂×#11715897	C₂³	1	1	yes	characteristic
46	C₂×#11715897	C₂³	1	1	yes	characteristic
47	C₂×#11715897	C₂³	1	1	yes	characteristic
48	C₂×#11715897	C₂³	1	2	yes	—
49	C₂×#11715897	C₂³	1	2	yes	—
50	C₂×D₃₈₈	C₂×C₄	1	1	yes	characteristic
51	C₂×C₂²⋊D₉₇	C₂×C₄	1	2	yes	—
52	C₂²×Dic₉₇	C₂×C₄	1	1	yes	characteristic
53	C₂²×Dic₉₇	C₂³	1	1	yes	characteristic
54	C₂×C₄×D₉₇	C₂³	1	2	yes	—
55	C₄⋊Dic₉₇	C₂×C₄	1	2	yes	—
56	C₄⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic
57	C₄⋊Dic₉₇	C₂×C₄	1	2	yes	—
58	#13095490	C₂×C₄	1	2	yes	—
59	#13095488	C₂×C₄	1	2	yes	—
60	#13095488	C₂×C₄	1	2	yes	—
61	#13095488	C₂×C₄	1	2	yes	—
62	#13095488	C₂×C₄	1	2	yes	—
63	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
66	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
67	C₄×Dic₉₇	C₂³	1	2	yes	—
68	C₂×Q₈×C₉₇	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
70	C₄⋊C₄×C₉₇	C₂×C₄	1	2	yes	—
71	C₄⋊C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
72	C₂²⋊C₄×C₉₇	C₂×C₄	1	2	yes	—
73	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×D₉₇	C₂²×C₄	1	2	yes	—
75	Q₈×C₉₇	—	2	2	no	—
76	C₂×C₄×C₉₇	—	2	2	no	—
77	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
81	C₂²⋊D₉₇	—	2	4	no	—
82	C₂×Dic₉₇	—	2	2	no	—
83	C₂×Dic₉₇	—	2	1	no	—
84	C₂×Dic₉₇	—	2	1	no	—
85	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
88	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
89	C₂×Dic₉₇	C₂⁴	1	1	yes	characteristic
90	D₃₈₈	—	2	2	no	—
91	C₄×D₉₇	—	2	2	no	—
92	C₄×D₉₇	—	2	2	no	—
93	#11715897	—	2	2	no	—
94	#11715897	—	2	2	no	—
95	#11715897	—	2	1	no	—
96	#11715897	—	2	1	no	—
97	#11715897	—	2	1	no	—
98	#11715897	—	2	1	no	—
99	C₂²×C₉₇	—	2	1	no	—
100	C₂²×C₉₇	—	2	1	no	—
101	C₂²×C₉₇	#131	1	2	yes	—
102	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
103	C₂×D₉₇	—	2	2	no	—
104	C₂×D₉₇	—	2	2	no	—
105	C₂×D₉₇	—	2	2	no	—
106	C₄×C₉₇	—	2	2	no	—
107	C₄×C₉₇	—	2	1	no	—
108	C₄×C₉₇	—	2	2	no	—
109	C₄×C₉₇	#131	1	2	yes	—
110	Dic₉₇	—	2	1	no	—
111	Dic₉₇	—	2	1	no	—
112	Dic₉₇	—	2	1	no	—
113	Dic₉₇	—	2	2	no	—
114	Dic₉₇	H(5,𝔽₂)	1	1	yes	characteristic
115	Dic₉₇	2₋¹⁺⁴	1	1	yes	characteristic
116	C₂×C₉₇	—	2	2	no	—
117	C₂×C₉₇	#520	1	1	yes	characteristic
118	C₂×C₉₇	C₂×#131	1	1	yes	characteristic
119	C₂×C₉₇	C₂×#131	1	1	yes	characteristic
120	D₉₇	—	4	2	no	—
121	#2994	—	97	1	no	maximal, Sylow
122	C₉₇	#2994	1	1	yes	characteristic, complemented, Sylow
123	#529	—	97	1	no	—
124	#446	—	97	1	no	—
125	C₈×Q₈	—	97	1	no	—
126	#436	—	97	1	no	—
127	#435	—	97	2	no	—
128	C₈×D₄	—	97	2	no	—
129	C₈×D₄	—	97	1	no	—
130	#433	—	97	2	no	—
131	#431	—	97	1	no	—
132	#424	—	97	2	no	—
133	C₂×C₄⋊C₈	—	97	1	no	—
134	C₂×M₁₆	—	97	2	no	—
135	C₂²×C₈	—	97	2	no	—
136	C₂²×C₈	—	97	2	no	—
137	#126	—	97	2	no	—
138	#125	—	97	1	no	—
139	#124	—	97	1	no	—
140	#123	—	97	2	no	—
141	C₂²⋊Q₈	#87604	1	1	yes	characteristic
142	#121	—	97	1	no	—
143	C₄×Q₈	—	97	1	no	—
144	C₄×D₄	—	97	1	no	—
145	C₄×D₄	—	97	2	no	—
146	#117	—	97	2	no	—
147	C₂×C₄²	—	97	1	no	—
148	C₄⋊C₈	—	97	2	no	—
149	C₄⋊C₈	—	97	1	no	—
150	C₄⋊C₈	—	97	1	no	—
151	C₄⋊C₈	—	97	2	no	—
152	C₄⋊C₈	—	97	1	no	—
153	C₄⋊C₈	—	97	2	no	—
154	C₄⋊C₈	—	97	1	no	—
155	C₂²⋊C₈	—	97	1	no	—
156	C₂²⋊C₈	—	97	1	no	—
157	C₂²⋊C₈	—	97	2	no	—
158	C₂²⋊C₈	—	97	2	no	—
159	#97	—	97	2	no	—
160	C₄×C₈	—	97	1	no	—
161	C₄×C₈	—	97	1	no	—
162	C₄×C₈	—	97	2	no	—
163	C₂×Q₈	C₂×#87604	1	1	yes	characteristic
164	C₂×D₄	—	97	2	no	—
165	C₂×D₄	—	97	1	no	—
166	C₂²×C₄	—	97	1	no	—
167	C₂²×C₄	—	97	1	no	—
168	C₂²×C₄	—	97	2	no	—
169	C₂²×C₄	C₂×#87604	1	1	yes	characteristic
170	M₁₆	—	388	2	no	—
171	C₂×C₈	—	97	1	no	—
172	C₂×C₈	—	97	1	no	—
173	C₂×C₈	—	97	1	no	—
174	C₂×C₈	—	97	2	no	—
175	C₂×C₈	—	97	1	no	—
176	C₂×C₈	—	97	2	no	—
177	C₂×C₈	—	388	2	no	—
178	C₂×C₈	—	388	2	no	—
179	C₄⋊C₄	—	97	2	no	—
180	C₄⋊C₄	—	97	2	no	—
181	C₄⋊C₄	—	97	1	no	—
182	C₄⋊C₄	—	97	2	no	—
183	C₄⋊C₄	C₂×#87604	1	2	yes	—
184	C₄⋊C₄	C₂×#87604	1	1	yes	characteristic
185	C₂²⋊C₄	—	97	2	no	—
186	C₂²⋊C₄	—	97	2	no	—
187	C₂²⋊C₄	—	97	2	no	—
188	C₂²⋊C₄	—	97	2	no	—
189	C₂²⋊C₄	C₂×#87604	1	2	yes	—
190	C₄²	—	97	1	no	—
191	C₄²	—	97	1	no	—
192	C₄²	—	97	1	no	—
193	C₄²	—	97	2	no	—
194	C₄²	—	97	1	no	—
195	C₂³	—	97	2	no	—
196	C₂³	C₂²×#87604	1	1	yes	characteristic
197	Q₈	—	2	2	no	—
198	D₄	—	194	4	no	—
199	D₄	—	194	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	97	1	no	—
202	C₂×C₄	—	97	1	no	—
203	C₂×C₄	—	97	1	no	—
204	C₂×C₄	—	97	1	no	—
205	C₂×C₄	—	97	2	no	—
206	C₂×C₄	—	194	2	no	—
207	C₂×C₄	—	194	2	no	—
208	C₂×C₄	—	194	1	no	—
209	C₂×C₄	—	194	1	no	—
210	C₂×C₄	—	194	2	no	—
211	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
212	C₂×C₄	C₂²×#87604	1	2	yes	—
213	C₂×C₄	C₂²×#87604	1	2	yes	—
214	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
215	C₈	—	194	2	no	—
216	C₈	—	194	1	no	—
217	C₈	—	194	2	no	—
218	C₈	—	194	1	no	—
219	C₈	—	194	1	no	—
220	C₈	—	194	1	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	194	2	no	—
224	C₂²	—	194	2	no	—
225	C₂²	—	194	2	no	—
226	C₂²	#15690898	1	2	yes	—
227	C₂²	C₂³×#87604	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	2	no	—
231	C₄	—	97	1	no	—
232	C₄	—	97	1	no	—
233	C₄	—	194	1	no	—
234	C₄	—	194	1	no	—
235	C₄	—	194	1	no	—
236	C₄	—	194	2	no	—
237	C₄	#15690897	1	2	yes	—
238	C₂	—	2	2	no	—
239	C₂	—	388	2	no	—
240	C₂	#20261804	1	1	yes	characteristic
241	C₂	C₂×#15690897	1	1	yes	characteristic
242	C₂	C₂×#15690898	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#87604
commutator, socle	C₂²×C₉₇	C₂³×C₄
Fitting	C₂²⋊Q₈×C₉₇	C₄
prime radical	#15690784	C₄
cosocle	C₂×Dic₉₇	C₂⁴
hypercenter	C₂²⋊Q₈	#87604
hypocenter	C₉₇	#2994
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #20261786	C₂	semidirect
2. Q₈×#11715897	C₂	semidirect
3. #20261769	C₂	semidirect
4. #20261764	C₂	semidirect
5. #20261763	C₂	semidirect
6. #20261762	C₂	semidirect
7. #20261760	C₂	semidirect
8. #20261752	C₂	semidirect
9. #20261742	C₂	semidirect
10. C₂×C₄⋊#11715897	C₂	semidirect
122. C₉₇	#2994	semidirect

Maximal subgroups

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

subgroup	index
#20261786	2
Q₈×#11715897	2
#20261769	2
#20261764	2
#20261763	2
#20261762	2
#20261760	2
#20261752	2
#20261742	2
C₂×C₄⋊#11715897	2
#20261271	2
#2994	97