Subgroups — C₂²×#16863523 #31720185

Order 17152.

2222 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	C₂²×C₄×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863523	C₂	1	24	yes	maximal, direct product factor
3	C₂²×Dic₆₇⋊C₈	C₂	1	2	yes	maximal
4	C₂²×#14523953	C₂	1	2	yes	maximal
5	C₂²×C₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₃₄	C₄	1	1	yes	characteristic
8	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₃₄	C₄	1	12	yes	—
11	C₂×C₄×Dic₁₃₄	C₂²	1	12	yes	—
12	C₂²×C₄⋊Dic₆₇	C₄	1	1	yes	characteristic
13	C₂²×Dic₆₇⋊C₄	C₄	1	2	yes	—
14	C₂²×C₄×Dic₆₇	C₂²	1	2	yes	—
15	#16863523	C₂²	1	64	yes	direct product factor
16	C₂×Dic₆₇⋊C₈	C₂²	1	24	yes	—
17	C₂×#14523953	C₂²	1	24	yes	—
18	C₂×C₄×C₆₇⋊C₈	C₂²	1	12	yes	—
19	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
20	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₈×C₆₇	C₂²	1	12	yes	—
22	C₂²×Dic₁₃₄	—	2	8	no	—
23	C₂²×Dic₁₃₄	C₂×C₄	1	12	yes	—
24	C₂³×Dic₆₇	C₂×C₄	1	2	yes	—
25	C₂³×Dic₆₇	C₂×C₄	1	2	yes	—
26	C₂²×C₆₇⋊C₈	—	2	4	no	—
27	C₂²×C₆₇⋊C₈	Q₈	1	8	yes	complemented
28	C₂²×C₆₇⋊C₈	C₂³	1	6	yes	—
29	C₂²×C₆₇⋊C₈	C₂³	1	6	yes	—
30	C₄×Dic₁₃₄	C₂×C₄	1	48	yes	—
31	C₄×Dic₁₃₄	C₂³	1	16	yes	—
32	C₂×C₄⋊Dic₆₇	C₂×C₄	1	12	yes	—
33	C₂×Dic₆₇⋊C₄	C₂×C₄	1	24	yes	—
34	C₂×C₄×Dic₆₇	C₂×C₄	1	12	yes	—
35	C₂×C₄×Dic₆₇	C₂³	1	12	yes	—
36	Dic₆₇⋊C₈	C₂³	1	32	yes	—
37	#14523953	C₂³	1	32	yes	—
38	C₄×C₆₇⋊C₈	C₂³	1	16	yes	—
39	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₆₇	—	2	8	no	—
43	C₂²×C₈×C₆₇	C₂³	1	12	yes	—
44	C₂×C₄²×C₆₇	C₂×C₄	1	6	yes	—
45	C₂×C₄²×C₆₇	C₂³	1	6	yes	—
46	C₄⋊C₈×C₆₇	C₂³	1	16	yes	—
47	C₂×Dic₁₃₄	—	2	48	no	—
48	C₂×Dic₁₃₄	C₂²×C₄	1	16	yes	—
49	C₂²×Dic₆₇	—	2	8	no	—
50	C₂²×Dic₆₇	—	2	8	no	—
51	C₂²×Dic₆₇	C₂²×C₄	1	12	yes	—
52	C₂²×Dic₆₇	C₂²×C₄	1	12	yes	—
53	C₂×C₆₇⋊C₈	—	2	12	no	—
54	C₂×C₆₇⋊C₈	C₂×Q₈	1	24	yes	complemented
55	C₂×C₆₇⋊C₈	C₂⁴	1	4	yes	—
56	C₂×C₆₇⋊C₈	C₂⁴	1	4	yes	—
57	C₄⋊Dic₆₇	C₂²×C₄	1	16	yes	—
58	Dic₆₇⋊C₄	C₂²×C₄	1	32	yes	—
59	C₄×Dic₆₇	C₂²×C₄	1	24	yes	—
60	C₄×Dic₆₇	C₂⁴	1	8	yes	—
61	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	—	2	4	no	—
63	C₂²×C₄×C₆₇	M₁₆	1	8	yes	—
64	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
65	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
66	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
67	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
68	C₂²×C₄×C₆₇	#41	1	4	yes	—
69	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
70	C₂×C₈×C₆₇	—	2	24	no	—
71	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
72	C₄²×C₆₇	C₂²×C₄	1	12	yes	—
73	C₄²×C₆₇	C₂⁴	1	4	yes	—
74	C₂³×C₆₇	C₄×Q₈	1	4	yes	—
75	C₂³×C₆₇	C₂×M₁₆	1	4	yes	—
76	C₂³×C₆₇	#131	1	4	yes	—
77	C₂³×C₆₇	C₂³×C₄	1	3	yes	—
78	C₂×C₄×C₆₇	—	2	12	no	—
79	C₂×C₄×C₆₇	C₄×Q₈	1	12	yes	—
80	C₂×C₄×C₆₇	C₂×M₁₆	1	24	yes	—
81	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
82	C₂×C₄×C₆₇	C₂³×C₄	1	3	yes	—
83	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
84	C₂×C₄×C₆₇	C₂²×Q₈	1	6	yes	—
85	C₂×C₄×C₆₇	C₂×#41	1	6	yes	—
86	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
87	C₂×Dic₆₇	—	2	24	no	—
88	C₂×Dic₆₇	—	2	24	no	—
89	C₂×Dic₆₇	C₂³×C₄	1	8	yes	—
90	C₂×Dic₆₇	C₂³×C₄	1	8	yes	—
91	Dic₁₃₄	—	2	32	no	—
92	C₈×C₆₇	—	2	8	no	—
93	C₆₇⋊C₈	—	2	4	no	—
94	C₆₇⋊C₈	C₂²×Q₈	1	8	yes	complemented
95	C₂²×C₆₇	#446	1	16	yes	complemented
96	C₂²×C₆₇	C₂×C₄×Q₈	1	6	yes	—
97	C₂²×C₆₇	C₂²×M₁₆	1	6	yes	—
98	C₂²×C₆₇	C₂×#131	1	6	yes	—
99	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
100	C₄×C₆₇	—	2	4	no	—
101	C₄×C₆₇	C₂×C₄×Q₈	1	6	yes	—
102	C₄×C₆₇	C₂²×M₁₆	1	8	yes	—
103	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
104	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
105	Dic₆₇	—	2	8	no	—
106	Dic₆₇	—	2	8	no	—
107	C₂²×#446	—	67	1	no	maximal, Sylow
108	C₂×C₆₇	C₂×#446	1	12	yes	complemented
109	C₂×C₆₇	C₂²×C₄×Q₈	1	1	yes	characteristic
110	C₂×C₆₇	C₂³×M₁₆	1	1	yes	characteristic
111	C₂×C₆₇	C₂²×#131	1	1	yes	characteristic
112	C₂²×C₄×Q₈	—	67	1	no	—
113	C₂×#446	—	67	24	no	—
114	C₂²×C₄⋊C₈	—	67	2	no	—
115	C₂²×C₄⋊C₈	D₆₇	1	1	yes	characteristic
116	C₂²×#97	—	67	2	no	—
117	C₂²×C₄×C₈	—	67	1	no	—
118	C₆₇	C₂²×#446	1	1	yes	characteristic, complemented, Sylow
119	C₂³×Q₈	—	67	1	no	—
120	C₂³×C₈	—	67	1	no	—
121	C₂³×C₈	—	67	1	no	—
122	C₂³×C₈	C₂×D₆₇	1	2	yes	—
123	C₂×C₄×Q₈	—	67	12	no	—
124	C₂×C₄×Q₈	—	67	12	no	—
125	C₂²×C₄⋊C₄	—	67	2	no	—
126	C₂²×C₄⋊C₄	—	67	1	no	—
127	C₂²×C₄²	—	67	2	no	—
128	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
129	#446	—	67	64	no	—
130	C₂×C₄⋊C₈	—	67	24	no	—
131	C₂×C₄⋊C₈	C₂×D₆₇	1	12	yes	—
132	C₂×#97	—	67	24	no	—
133	C₂×C₄×C₈	—	67	12	no	—
134	C₂²×Q₈	—	67	12	no	—
135	C₂²×Q₈	—	134	8	no	—
136	C₂³×C₄	—	67	2	no	—
137	C₂³×C₄	—	67	2	no	—
138	C₂³×C₄	C₄×D₆₇	1	1	yes	characteristic
139	C₂³×C₄	C₄×D₆₇	1	1	yes	characteristic
140	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
141	C₂²×C₈	—	2	8	no	—
142	C₂²×C₈	—	67	8	no	—
143	C₂²×C₈	—	67	6	no	—
144	C₂²×C₈	—	67	6	no	—
145	C₂²×C₈	—	134	4	no	—
146	C₂²×C₈	C₂²×D₆₇	1	12	yes	—
147	C₄×Q₈	—	67	48	no	—
148	C₄×Q₈	—	67	16	no	—
149	C₂×C₄⋊C₄	—	67	24	no	—
150	C₂×C₄⋊C₄	—	67	12	no	—
151	C₂×C₄²	—	67	12	no	—
152	C₂×C₄²	—	67	12	no	—
153	C₂×C₄²	C₄×D₆₇	1	6	yes	—
154	C₂×C₄²	C₂²×D₆₇	1	6	yes	—
155	C₄⋊C₈	—	67	32	no	—
156	C₄⋊C₈	C₂²×D₆₇	1	16	yes	—
157	#97	—	67	32	no	—
158	C₄×C₈	—	67	16	no	—
159	C₂⁴	C₂×C₄×D₆₇	1	1	yes	characteristic
160	C₂×Q₈	—	67	16	no	—
161	C₂×Q₈	—	134	48	no	—
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	67	12	no	—
164	C₂²×C₄	—	67	12	no	—
165	C₂²×C₄	—	134	8	no	—
166	C₂²×C₄	—	134	8	no	—
167	C₂²×C₄	#11761611	1	8	yes	—
168	C₂²×C₄	C₂×C₄×D₆₇	1	6	yes	—
169	C₂²×C₄	C₂×C₄×D₆₇	1	3	yes	—
170	C₂²×C₄	C₂×C₄×D₆₇	1	6	yes	—
171	C₂²×C₄	Q₈×D₆₇	1	4	yes	—
172	C₂²×C₄	#11761627	1	4	yes	—
173	C₂²×C₄	C₂³×D₆₇	1	3	yes	—
174	C₂×C₈	—	2	24	no	—
175	C₂×C₈	—	67	24	no	—
176	C₂×C₈	—	67	4	no	—
177	C₂×C₈	—	67	4	no	—
178	C₂×C₈	—	134	12	no	—
179	C₂×C₈	C₂³×D₆₇	1	8	yes	—
180	C₄⋊C₄	—	67	32	no	—
181	C₄⋊C₄	—	67	16	no	—
182	C₄²	—	67	24	no	—
183	C₄²	—	67	8	no	—
184	C₄²	C₂×C₄×D₆₇	1	12	yes	—
185	C₄²	C₂³×D₆₇	1	4	yes	—
186	C₂³	#14524008	1	4	yes	—
187	C₂³	C₂×#11761611	1	4	yes	—
188	C₂³	#14524051	1	4	yes	—
189	C₂³	C₂²×C₄×D₆₇	1	3	yes	—
190	Q₈	—	134	32	no	—
191	C₂×C₄	—	2	12	no	—
192	C₂×C₄	—	67	8	no	—
193	C₂×C₄	—	67	8	no	—
194	C₂×C₄	—	134	24	no	—
195	C₂×C₄	—	134	24	no	—
196	C₂×C₄	#14524008	1	12	yes	—
197	C₂×C₄	C₂×#11761611	1	24	yes	—
198	C₂×C₄	C₂²×C₄×D₆₇	1	4	yes	—
199	C₂×C₄	C₂²×C₄×D₆₇	1	3	yes	—
200	C₂×C₄	C₂²×C₄×D₆₇	1	4	yes	—
201	C₂×C₄	C₂×Q₈×D₆₇	1	6	yes	—
202	C₂×C₄	C₂×#11761627	1	6	yes	—
203	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
204	C₈	—	2	8	no	—
205	C₈	—	67	8	no	—
206	C₈	—	134	4	no	—
207	C₂²	#16863523	1	16	yes	direct product factor
208	C₂²	C₂×#14524008	1	6	yes	—
209	C₂²	C₂²×#11761611	1	6	yes	—
210	C₂²	C₂×#14524051	1	6	yes	—
211	C₂²	C₂³×C₄×D₆₇	1	1	yes	characteristic
212	C₄	—	2	4	no	—
213	C₄	—	134	8	no	—
214	C₄	—	134	8	no	—
215	C₄	C₂×#14524008	1	6	yes	—
216	C₄	C₂²×#11761611	1	8	yes	—
217	C₄	C₂²×Q₈×D₆₇	1	1	yes	characteristic
218	C₄	C₂²×#11761627	1	1	yes	characteristic
219	C₂	C₂×#16863523	1	12	yes	direct product factor
220	C₂	C₂²×#14524008	1	1	yes	characteristic
221	C₂	C₂³×#11761611	1	1	yes	characteristic
222	C₂	C₂²×#14524051	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂²×C₄⋊C₈×C₆₇	C₂
socle, prime radical	C₂⁴×C₆₇	C₂²×C₄
cosocle	C₂×C₄×C₆₇	C₂⁵
hypercenter	C₂²×C₄⋊C₈	D₆₇
hypocenter	C₆₇	C₂²×#446
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#16863523	C₂	direct (N × H)
15. #16863523	C₂²	direct (N × H)
27. C₂²×C₆₇⋊C₈	Q₈	semidirect
54. C₂×C₆₇⋊C₈	C₂×Q₈	semidirect
94. C₆₇⋊C₈	C₂²×Q₈	semidirect
95. C₂²×C₆₇	#446	semidirect
108. C₂×C₆₇	C₂×#446	semidirect
118. C₆₇	C₂²×#446	semidirect
207. C₂²	#16863523	direct (N × H)
219. C₂	C₂×#16863523	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×Dic₁₃₄	2
C₂×#16863523	2
C₂²×Dic₆₇⋊C₈	2
C₂²×#14523953	2
C₂²×C₄×C₆₇⋊C₈	2
C₂²×C₄⋊C₈×C₆₇	2
C₂²×#446	67

Subgroups — C22×#16863523 #31720185

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16863523 #31720185