Subgroups — #30770889

Order 17152.

510 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	#22561820	C₂	1	2	yes	maximal
2	C₂×#16863174	C₂	1	2	yes	maximal, complemented
3	#22558289	C₂	1	2	yes	maximal, complemented
4	C₂²×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523983	C₂²	1	4	yes	—
6	C₂²⋊(C₆₇⋊C₁₆)	C₄	1	8	yes	complemented
7	#16863174	—	2	4	no	—
8	C₂³×C₈×C₆₇	C₄	1	1	yes	characteristic
9	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₆₇	C₄	1	1	yes	characteristic
11	C₂×#97×C₆₇	—	2	4	no	—
12	C₂×#97×C₆₇	C₄	1	1	yes	characteristic
13	C₂×#97×C₆₇	C₄	1	1	yes	characteristic
14	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
16	#14523983	—	2	4	no	—
17	#14523983	—	2	4	no	—
18	C₂×C₆₇⋊C₁₆	C₂×C₄	1	8	yes	complemented
19	C₂³×C₄×C₆₇	—	2	1	no	—
20	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₆₇	—	2	2	no	—
22	C₂²×C₈×C₆₇	—	2	2	no	—
23	C₂²×C₈×C₆₇	—	2	2	no	—
24	C₂²×C₈×C₆₇	—	2	2	no	—
25	C₂²×C₈×C₆₇	—	2	2	no	—
26	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₆₇	—	2	1	no	—
32	C₂×C₄²×C₆₇	—	2	2	no	—
33	C₂×C₄²×C₆₇	—	2	2	no	—
34	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
36	#97×C₆₇	—	2	2	no	—
37	#97×C₆₇	—	2	2	no	—
38	#97×C₆₇	—	2	2	no	—
39	#97×C₆₇	D₄	1	4	yes	—
40	C₆₇⋊C₁₆	—	2	8	no	—
41	C₂⁴×C₆₇	M₁₆	1	1	yes	characteristic
42	C₂²×C₄×C₆₇	—	2	1	no	—
43	C₂²×C₄×C₆₇	—	2	1	no	—
44	C₂²×C₄×C₆₇	—	2	1	no	—
45	C₂²×C₄×C₆₇	—	2	1	no	—
46	C₂²×C₄×C₆₇	—	2	4	no	—
47	C₂²×C₄×C₆₇	—	2	1	no	—
48	C₂²×C₄×C₆₇	—	2	1	no	—
49	C₂²×C₄×C₆₇	—	4	4	no	—
50	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
51	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	M₁₆	1	2	yes	—
53	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₆₇	—	2	8	no	—
55	C₂×C₈×C₆₇	—	2	1	no	—
56	C₂×C₈×C₆₇	—	2	1	no	—
57	C₂×C₈×C₆₇	—	2	1	no	—
58	C₂×C₈×C₆₇	—	2	1	no	—
59	C₂×C₈×C₆₇	—	2	1	no	—
60	C₂×C₈×C₆₇	—	2	1	no	—
61	C₂×C₈×C₆₇	—	2	8	no	—
62	C₂×C₈×C₆₇	C₄²	1	2	yes	—
63	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
65	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
66	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₆₇	#41	1	2	yes	—
68	C₄²×C₆₇	—	2	4	no	—
69	C₄²×C₆₇	—	2	1	no	—
70	C₄²×C₆₇	—	2	1	no	—
71	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
72	C₄²×C₆₇	M₁₆	1	2	yes	—
73	C₂³×C₆₇	—	2	4	no	—
74	C₂³×C₆₇	—	2	1	no	—
75	C₂³×C₆₇	—	2	1	no	—
76	C₂³×C₆₇	#97	1	2	yes	—
77	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₆₇	—	2	1	no	—
79	C₂×C₄×C₆₇	—	2	1	no	—
80	C₂×C₄×C₆₇	—	2	2	no	—
81	C₂×C₄×C₆₇	—	2	2	no	—
82	C₂×C₄×C₆₇	—	2	2	no	—
83	C₂×C₄×C₆₇	—	2	2	no	—
84	C₂×C₄×C₆₇	—	2	2	no	—
85	C₂×C₄×C₆₇	—	2	2	no	—
86	C₂×C₄×C₆₇	—	4	8	no	—
87	C₂×C₄×C₆₇	—	4	2	no	—
88	C₂×C₄×C₆₇	—	4	2	no	—
89	C₂×C₄×C₆₇	#97	1	2	yes	—
90	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
91	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
95	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
96	C₈×C₆₇	—	2	2	no	—
97	C₈×C₆₇	—	2	2	no	—
98	C₈×C₆₇	—	2	2	no	—
99	C₈×C₆₇	C₄×D₄	1	4	yes	—
100	C₂²×C₆₇	—	2	4	no	—
101	C₂²×C₆₇	—	2	4	no	—
102	C₂²×C₆₇	—	2	2	no	—
103	C₂²×C₆₇	—	2	2	no	—
104	C₂²×C₆₇	—	2	2	no	—
105	C₂²×C₆₇	#347	1	4	yes	complemented
106	C₂²×C₆₇	C₂×#97	1	1	yes	characteristic
107	C₂²×C₆₇	#407	1	1	yes	characteristic
108	C₂²×C₆₇	#432	1	1	yes	characteristic
109	C₄×C₆₇	—	2	2	no	—
110	C₄×C₆₇	—	4	2	no	—
111	C₄×C₆₇	—	4	2	no	—
112	C₄×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
113	C₄×C₆₇	C₄×M₁₆	1	1	yes	characteristic
114	C₄×C₆₇	#407	1	1	yes	characteristic
115	C₄×C₆₇	#432	1	1	yes	characteristic
116	#12605	—	67	1	no	maximal, Sylow
117	C₂×C₆₇	—	2	1	no	—
118	C₂×C₆₇	—	2	4	no	—
119	C₂×C₆₇	—	2	1	no	—
120	C₂×C₆₇	#1749	1	1	yes	characteristic
121	C₂×C₆₇	C₂×#347	1	2	yes	—
122	C₂²×#97	D₆₇	1	1	yes	characteristic
123	#2112	—	67	2	no	—
124	C₂×#347	—	67	2	no	—
125	#1375	—	67	2	no	—
126	C₆₇	#12605	1	1	yes	characteristic, complemented, Sylow
127	C₂³×C₈	Dic₆₇	1	1	yes	characteristic
128	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
129	C₂²×C₄²	Dic₆₇	1	1	yes	characteristic
130	C₂×M₃₂	—	67	4	no	—
131	C₂×#97	—	2	4	no	—
132	C₂×#97	Dic₆₇	1	1	yes	characteristic
133	C₂×#97	Dic₆₇	1	1	yes	characteristic
134	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
135	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
136	C₂²⋊C₁₆	—	67	8	no	—
137	#347	—	134	4	no	—
138	C₂³×C₄	—	2	1	no	—
139	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
140	C₂²×C₈	—	2	2	no	—
141	C₂²×C₈	—	2	2	no	—
142	C₂²×C₈	—	2	2	no	—
143	C₂²×C₈	—	2	2	no	—
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	C₄×D₆₇	1	4	yes	—
146	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
147	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
148	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
149	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
150	C₂×C₄²	—	2	1	no	—
151	C₂×C₄²	—	2	2	no	—
152	C₂×C₄²	—	2	2	no	—
153	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
154	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
155	M₃₂	—	134	4	no	—
156	M₃₂	—	134	4	no	—
157	C₂×C₁₆	—	67	8	no	—
158	#97	—	2	2	no	—
159	#97	—	2	2	no	—
160	#97	—	2	2	no	—
161	#97	C₂²⋊D₆₇	1	4	yes	—
162	C₂⁴	#11761612	1	1	yes	characteristic
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	2	4	no	—
168	C₂²×C₄	—	2	1	no	—
169	C₂²×C₄	—	2	1	no	—
170	C₂²×C₄	—	4	4	no	—
171	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
172	C₂²×C₄	#11761612	1	1	yes	characteristic
173	C₂²×C₄	#11761612	1	2	yes	—
174	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
175	C₂×C₈	—	2	8	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	8	no	—
183	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
184	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
185	C₂×C₈	#11761612	1	2	yes	—
186	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
187	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
188	C₂×C₈	#11761628	1	2	yes	—
189	C₄²	—	2	4	no	—
190	C₄²	—	2	1	no	—
191	C₄²	—	2	1	no	—
192	C₄²	C₂²⋊Dic₆₇	1	2	yes	—
193	C₄²	#11761612	1	2	yes	—
194	C₁₆	—	134	8	no	—
195	C₂³	—	2	1	no	—
196	C₂³	—	2	4	no	—
197	C₂³	—	2	1	no	—
198	C₂³	#14523954	1	2	yes	—
199	C₂³	C₂×#11761612	1	1	yes	characteristic
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	4	8	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	#14523954	1	2	yes	—
212	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
213	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
214	C₂×C₄	#14524002	1	1	yes	characteristic
215	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
216	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
217	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
218	C₈	—	2	2	no	—
219	C₈	—	2	2	no	—
220	C₈	—	2	2	no	—
221	C₈	C₄×C₂²⋊D₆₇	1	4	yes	—
222	C₂²	—	2	4	no	—
223	C₂²	—	2	4	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	2	no	—
227	C₂²	#16863174	1	4	yes	complemented
228	C₂²	C₂×#14523954	1	1	yes	characteristic
229	C₂²	#16863360	1	1	yes	characteristic
230	C₂²	#16863453	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	4	2	no	—
233	C₄	—	4	2	no	—
234	C₄	C₄×C₂²⋊Dic₆₇	1	1	yes	characteristic
235	C₄	C₄×#11761612	1	1	yes	characteristic
236	C₄	#16863360	1	1	yes	characteristic
237	C₄	#16863453	1	1	yes	characteristic
238	C₂	—	2	1	no	—
239	C₂	—	2	4	no	—
240	C₂	—	2	1	no	—
241	C₂	#22559971	1	1	yes	characteristic
242	C₂	C₂×#16863174	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₆₇
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂²×#97×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#97
prime radical	C₂⁴×C₆₇	M₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂²×#97	D₆₇
hypocenter	C₆₇	#12605
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
2. C₂×#16863174	C₂	semidirect
3. #22558289	C₂	semidirect
6. C₂²⋊(C₆₇⋊C₁₆)	C₄	semidirect
18. C₂×C₆₇⋊C₁₆	C₂×C₄	semidirect
105. C₂²×C₆₇	#347	semidirect
126. C₆₇	#12605	semidirect
227. C₂²	#16863174	semidirect

Maximal subgroups

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

subgroup	index
#22561820	2
C₂×#16863174	2
#22558289	2
C₂²×#97×C₆₇	2
#12605	67