Subgroups — #27158175

Order 12416.

780 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×#87604	C₂	1	2	yes	maximal
2	C₄⋊C₄×#87604	C₂	1	2	yes	maximal, complemented
3	C₄×C₄⋊#87604	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²⋊C₄×#87604	C₂	1	4	yes	maximal, complemented
5	C₄×C₂²⋊#87604	C₂	1	2	yes	maximal, complemented
6	#20261671	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×(C₂×#87604)⋊C₄	C₂	1	2	yes	maximal, complemented
8	C₄×D₄×D₉₇	C₂	1	1	yes	maximal, characteristic
9	C₂³×#87604	C₂²	1	4	yes	—
10	D₄×#87604	C₄	1	16	yes	complemented
11	C₂×C₄⋊#87604	C₂²	1	2	yes	—
12	C₂×C₂²⋊#87604	C₂²	1	4	yes	—
13	C₂×C₄×#87604	C₂²	1	4	yes	—
14	C₂×C₄×#87604	C₂²	1	2	yes	—
15	C₂×C₄×#87604	C₂²	1	2	yes	—
16	(C₂×#87604)⋊C₄	C₂²	1	1	yes	characteristic
17	(C₂×#87604)⋊C₄	C₂²	1	2	yes	—
18	(C₂×#87604)⋊C₄	C₂²	1	4	yes	—
19	(C₂×#87604)⋊C₄	C₂²	1	1	yes	characteristic
20	(C₂×#87604)⋊C₄	C₂²	1	2	yes	—
21	C₂×D₄×D₉₇	C₂²	1	1	yes	characteristic
22	C₂²×C₄×D₉₇	C₂²	1	2	yes	—
23	C₄×D₃₈₈	C₄	1	1	yes	characteristic, complemented
24	D₄×Dic₉₇	C₄	1	1	yes	characteristic, complemented
25	#15690772	C₄	1	1	yes	characteristic, complemented
26	C₄×C₂²⋊D₉₇	C₄	1	2	yes	complemented
27	#15690770	C₄	1	2	yes	complemented
28	C₄⋊C₄×D₉₇	C₂²	1	1	yes	characteristic
29	C₂²⋊C₄×D₉₇	C₂²	1	2	yes	—
30	C₄²×D₉₇	C₂²	1	1	yes	characteristic
31	C₄×D₄×C₉₇	C₄	1	1	yes	characteristic, complemented
32	C₂²×#87604	—	4	4	no	—
33	C₂²×#87604	—	4	4	no	—
34	C₂²×#87604	C₂×C₄	1	16	yes	complemented
35	C₂²×#87604	C₂³	1	2	yes	—
36	C₂²×#87604	C₂³	1	2	yes	—
37	C₂²×#87604	C₂³	1	4	yes	—
38	C₄⋊#87604	C₂×C₄	1	8	yes	complemented
39	C₂²⋊#87604	C₂×C₄	1	16	yes	complemented
40	C₄×#87604	—	4	4	no	—
41	C₄×#87604	—	4	2	no	—
42	C₄×#87604	C₂×C₄	1	8	yes	complemented
43	C₂³×D₉₇	C₂³	1	2	yes	—
44	C₂×D₃₈₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₂²⋊D₉₇	C₂×C₄	1	2	yes	—
46	D₄×D₉₇	C₂×C₄	1	8	yes	—
47	C₂²×Dic₉₇	C₂×C₄	1	2	yes	complemented
48	C₂×C₄×D₉₇	—	4	4	no	—
49	C₂×C₄×D₉₇	C₂×C₄	1	2	yes	complemented
50	C₂×C₄×D₉₇	C₂×C₄	1	2	yes	complemented
51	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
52	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
53	C₂×C₄×D₉₇	C₂³	1	2	yes	—
54	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
55	C₄⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic, complemented
56	#13095490	C₂×C₄	1	2	yes	complemented
57	C₂²⋊Dic₉₇	C₂×C₄	1	2	yes	complemented
58	#13095488	C₂×C₄	1	2	yes	complemented
59	#13095488	C₂×C₄	1	2	yes	complemented
60	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic, complemented
61	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic, complemented
62	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic, complemented
63	C₂×D₄×C₉₇	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	complemented
65	C₄⋊C₄×C₉₇	C₂×C₄	1	1	yes	characteristic, complemented
66	C₂²⋊C₄×C₉₇	C₂×C₄	1	2	yes	complemented
67	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic, complemented
68	C₂³×C₉₇	C₂²×C₄	1	2	yes	—
69	C₂²×D₉₇	—	2	4	no	—
70	C₂²×D₉₇	—	2	2	no	—
71	C₂²×D₉₇	—	2	2	no	—
72	C₂²×D₉₇	C₂²×C₄	1	2	yes	—
73	C₂²×D₉₇	C₂²×C₄	1	8	yes	—
74	C₂²×D₉₇	C₂⁴	1	1	yes	characteristic
75	C₂×#87604	—	2	2	no	—
76	C₂×#87604	—	2	4	no	—
77	C₂×#87604	—	2	4	no	—
78	C₂×#87604	—	2	2	no	—
79	C₂×#87604	—	2	2	no	—
80	C₂×#87604	—	2	2	no	—
81	C₂×#87604	—	4	16	no	—
82	C₂×#87604	C₂²×C₄	1	4	yes	—
83	C₂×#87604	C₂²×C₄	1	4	yes	—
84	C₂×#87604	C₂²×C₄	1	8	yes	—
85	D₄×C₉₇	C₄²	1	4	yes	complemented
86	C₂×C₄×C₉₇	—	4	2	no	—
87	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
91	C₂²⋊D₉₇	C₄²	1	8	yes	complemented
92	C₂×Dic₉₇	—	4	2	no	—
93	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
96	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
97	D₃₈₈	C₄²	1	4	yes	complemented
98	C₄×D₉₇	—	2	2	no	—
99	C₄×D₉₇	—	2	4	no	—
100	C₄×D₉₇	—	2	1	no	—
101	C₄×D₉₇	—	2	1	no	—
102	C₄×D₉₇	—	4	2	no	—
103	C₄×D₉₇	—	4	2	no	—
104	C₄×D₉₇	C₂²×C₄	1	4	yes	—
105	C₂²×C₉₇	—	2	2	no	—
106	C₂²×C₉₇	—	2	2	no	—
107	C₂²×C₉₇	C₂×C₄²	1	4	yes	—
108	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
109	C₂×D₉₇	—	2	2	no	—
110	C₂×D₉₇	—	2	2	no	—
111	C₂×D₉₇	—	2	8	no	—
112	C₂×D₉₇	—	2	8	no	—
113	C₂×D₉₇	C₂×C₄²	1	4	yes	—
114	C₂×D₉₇	C₂³×C₄	1	2	yes	—
115	C₂×D₉₇	H(5,𝔽₂)	1	1	yes	characteristic
116	C₂×D₉₇	H(5,𝔽₂)	1	2	yes	—
117	C₂×D₉₇	2₋¹⁺⁴	1	1	yes	characteristic
118	#87604	—	2	4	no	—
119	#87604	—	2	8	no	—
120	#87604	—	2	4	no	—
121	C₄×C₉₇	—	2	1	no	—
122	C₄×C₉₇	—	2	2	no	—
123	C₄×C₉₇	—	2	1	no	—
124	C₄×C₉₇	C₂×C₄²	1	2	yes	—
125	Dic₉₇	—	2	1	no	—
126	Dic₉₇	—	2	2	no	—
127	Dic₉₇	—	2	1	no	—
128	Dic₉₇	C₂×C₄²	1	2	yes	—
129	C₂×C₉₇	—	2	4	no	—
130	C₂×C₉₇	C₂²×C₄²	1	1	yes	characteristic
131	C₂×C₉₇	#518	1	1	yes	characteristic
132	C₂×C₉₇	#520	1	1	yes	characteristic
133	D₉₇	—	2	4	no	—
134	D₉₇	#518	1	2	yes	—
135	D₉₇	#520	1	2	yes	—
136	#2275	—	97	1	no	maximal, Sylow
137	C₉₇	#2275	1	1	yes	characteristic, complemented, Sylow
138	C₂×C₄×D₄	—	97	2	no	—
139	C₂×C₄×D₄	—	97	1	no	—
140	C₄×C₄⋊C₄	—	97	1	no	—
141	C₄×C₄⋊C₄	—	97	2	no	—
142	C₄×C₂²⋊C₄	—	97	2	no	—
143	C₄×C₂²⋊C₄	—	97	4	no	—
144	#376	—	97	1	no	—
145	C₂×#95	—	97	2	no	—
146	C₂²×D₄	—	97	1	no	—
147	C₂³×C₄	—	97	4	no	—
148	C₂³×C₄	—	97	2	no	—
149	C₄×D₄	—	97	16	no	—
150	C₄×D₄	—	97	1	no	—
151	C₄×D₄	—	97	2	no	—
152	C₄×D₄	—	97	1	no	—
153	C₄×D₄	—	97	2	no	—
154	C₄×D₄	—	97	1	no	—
155	C₄×D₄	#87604	1	1	yes	characteristic, complemented
156	C₂×C₄⋊C₄	—	97	2	no	—
157	C₂×C₄⋊C₄	—	97	1	no	—
158	C₂×C₂²⋊C₄	—	97	4	no	—
159	C₂×C₂²⋊C₄	—	97	2	no	—
160	C₂×C₄²	—	97	2	no	—
161	C₂×C₄²	—	97	1	no	—
162	C₂×C₄²	—	97	4	no	—
163	C₂×C₄²	—	97	2	no	—
164	#95	—	97	1	no	—
165	#95	—	97	4	no	—
166	#95	—	97	2	no	—
167	#95	—	97	2	no	—
168	#95	—	97	1	no	—
169	C₂⁴	—	97	2	no	—
170	C₂×D₄	—	97	2	no	—
171	C₂×D₄	—	97	8	no	—
172	C₂×D₄	—	97	1	no	—
173	C₂×D₄	C₂×#87604	1	1	yes	characteristic
174	C₂²×C₄	—	97	1	no	—
175	C₂²×C₄	—	97	16	no	—
176	C₂²×C₄	—	97	2	no	—
177	C₂²×C₄	—	97	2	no	—
178	C₂²×C₄	—	97	2	no	—
179	C₂²×C₄	—	97	4	no	—
180	C₂²×C₄	—	97	2	no	—
181	C₂²×C₄	—	97	2	no	—
182	C₂²×C₄	—	97	1	no	—
183	C₂²×C₄	—	97	1	no	—
184	C₂²×C₄	—	97	2	no	—
185	C₂²×C₄	—	388	4	no	—
186	C₂²×C₄	—	388	4	no	—
187	C₂²×C₄	—	388	4	no	—
188	C₂²×C₄	C₂×#87604	1	2	yes	complemented
189	C₄⋊C₄	—	97	8	no	—
190	C₄⋊C₄	—	97	2	no	—
191	C₄⋊C₄	—	97	1	no	—
192	C₄⋊C₄	C₂×#87604	1	1	yes	characteristic, complemented
193	C₂²⋊C₄	—	97	16	no	—
194	C₂²⋊C₄	—	97	2	no	—
195	C₂²⋊C₄	—	97	2	no	—
196	C₂²⋊C₄	—	97	2	no	—
197	C₂²⋊C₄	C₂×#87604	1	2	yes	complemented
198	C₄²	—	97	8	no	—
199	C₄²	—	97	1	no	—
200	C₄²	—	97	1	no	—
201	C₄²	—	97	1	no	—
202	C₄²	—	388	2	no	—
203	C₄²	—	388	4	no	—
204	C₄²	C₂×#87604	1	1	yes	characteristic, complemented
205	C₂³	—	97	1	no	—
206	C₂³	—	97	8	no	—
207	C₂³	—	97	2	no	—
208	C₂³	—	194	4	no	—
209	C₂³	—	194	2	no	—
210	C₂³	—	194	2	no	—
211	C₂³	C₂²×#87604	1	2	yes	—
212	D₄	—	97	8	no	—
213	D₄	—	97	4	no	—
214	D₄	C₄×#87604	1	4	yes	complemented
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	97	1	no	—
217	C₂×C₄	—	97	4	no	—
218	C₂×C₄	—	97	4	no	—
219	C₂×C₄	—	97	4	no	—
220	C₂×C₄	—	97	8	no	—
221	C₂×C₄	—	97	1	no	—
222	C₂×C₄	—	97	1	no	—
223	C₂×C₄	—	97	2	no	—
224	C₂×C₄	—	388	2	no	—
225	C₂×C₄	—	388	2	no	—
226	C₂×C₄	—	388	2	no	—
227	C₂×C₄	—	388	16	no	—
228	C₂×C₄	—	194	1	no	—
229	C₂×C₄	—	194	4	no	—
230	C₂×C₄	—	194	2	no	—
231	C₂×C₄	—	194	1	no	—
232	C₂×C₄	—	194	2	no	—
233	C₂×C₄	—	194	4	no	—
234	C₂×C₄	—	194	4	no	—
235	C₂×C₄	—	194	2	no	—
236	C₂×C₄	—	194	2	no	—
237	C₂×C₄	—	194	2	no	—
238	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
239	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
240	C₂×C₄	C₂²×#87604	1	2	yes	—
241	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	97	2	no	—
245	C₂²	—	97	2	no	—
246	C₂²	—	97	1	no	—
247	C₂²	—	97	1	no	—
248	C₂²	—	97	4	no	—
249	C₂²	—	194	8	no	—
250	C₂²	—	194	8	no	—
251	C₂²	—	194	2	no	—
252	C₂²	—	194	2	no	—
253	C₂²	C₂×C₄×#87604	1	4	yes	—
254	C₂²	C₂³×#87604	1	1	yes	characteristic
255	C₄	—	2	1	no	—
256	C₄	—	2	2	no	—
257	C₄	—	2	1	no	—
258	C₄	—	97	2	no	—
259	C₄	—	194	4	no	—
260	C₄	—	194	8	no	—
261	C₄	—	194	4	no	—
262	C₄	—	194	1	no	—
263	C₄	—	194	2	no	—
264	C₄	—	194	1	no	—
265	C₄	C₂×C₄×#87604	1	2	yes	—
266	C₂	—	2	4	no	—
267	C₂	—	97	2	no	—
268	C₂	—	97	2	no	—
269	C₂	—	194	4	no	—
270	C₂	C₂²×C₄×#87604	1	1	yes	characteristic
271	C₂	#20261802	1	1	yes	characteristic
272	C₂	#20261804	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#87604
commutator	C₂×C₉₇	C₂²×C₄²
Fitting	C₄×D₄×C₉₇	C₄
socle	C₂²×C₉₇	C₂³×C₄
prime radical	C₂×D₄×D₉₇	C₂²
cosocle	C₂²×D₉₇	C₂⁴
hypercenter	C₄×D₄	#87604
hypocenter	C₉₇	#2275
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₄⋊C₄×#87604	C₂	semidirect
3. C₄×C₄⋊#87604	C₂	semidirect
4. C₂²⋊C₄×#87604	C₂	semidirect
5. C₄×C₂²⋊#87604	C₂	semidirect
6. #20261671	C₂	semidirect
7. C₂×(C₂×#87604)⋊C₄	C₂	semidirect
10. D₄×#87604	C₄	semidirect
23. C₄×D₃₈₈	C₄	semidirect
24. D₄×Dic₉₇	C₄	semidirect
25. #15690772	C₄	semidirect
26. C₄×C₂²⋊D₉₇	C₄	semidirect
27. #15690770	C₄	semidirect
31. C₄×D₄×C₉₇	C₄	semidirect
34. C₂²×#87604	C₂×C₄	semidirect
38. C₄⋊#87604	C₂×C₄	semidirect
39. C₂²⋊#87604	C₂×C₄	semidirect
42. C₄×#87604	C₂×C₄	semidirect
47. C₂²×Dic₉₇	C₂×C₄	semidirect
49. C₂×C₄×D₉₇	C₂×C₄	semidirect
50. C₂×C₄×D₉₇	C₂×C₄	semidirect
55. C₄⋊Dic₉₇	C₂×C₄	semidirect
56. #13095490	C₂×C₄	semidirect
57. C₂²⋊Dic₉₇	C₂×C₄	semidirect
58. #13095488	C₂×C₄	semidirect
59. #13095488	C₂×C₄	semidirect
60. C₄×Dic₉₇	C₂×C₄	semidirect
61. C₄×Dic₉₇	C₂×C₄	semidirect
62. C₄×Dic₉₇	C₂×C₄	semidirect
64. C₂²×C₄×C₉₇	C₂×C₄	semidirect
65. C₄⋊C₄×C₉₇	C₂×C₄	semidirect
66. C₂²⋊C₄×C₉₇	C₂×C₄	semidirect
67. C₄²×C₉₇	C₂×C₄	semidirect
85. D₄×C₉₇	C₄²	semidirect
91. C₂²⋊D₉₇	C₄²	semidirect
97. D₃₈₈	C₄²	semidirect
137. C₉₇	#2275	semidirect
155. C₄×D₄	#87604	semidirect
188. C₂²×C₄	C₂×#87604	semidirect
192. C₄⋊C₄	C₂×#87604	semidirect
197. C₂²⋊C₄	C₂×#87604	semidirect
204. C₄²	C₂×#87604	semidirect
214. D₄	C₄×#87604	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×#87604	2
C₄⋊C₄×#87604	2
C₄×C₄⋊#87604	2
C₂²⋊C₄×#87604	2
C₄×C₂²⋊#87604	2
#20261671	2
C₂×(C₂×#87604)⋊C₄	2
C₄×D₄×D₉₇	2
#2275	97