Subgroups — C₆₇×#14874 #30694125

Order 17152.

4118 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#395×C₆₇	C₂	1	4	yes	maximal, complemented
3	C₂×#385×C₆₇	C₂	1	3	yes	maximal
4	C₆₇×#1446	C₂	1	4	yes	maximal, complemented
5	C₆₇×C₂²⋊#95	C₂	1	3	yes	maximal
6	C₂³×Q₈×C₆₇	—	2	16	no	—
7	C₂³×Q₈×C₆₇	—	2	12	no	—
8	C₂³×Q₈×C₆₇	C₂²	1	4	yes	—
9	C₂⁴×C₄×C₆₇	C₂²	1	3	yes	—
10	C₂²×C₄⋊C₄×C₆₇	C₂²	1	6	yes	—
11	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
12	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	3	yes	—
13	#395×C₆₇	—	2	16	no	—
14	#385×C₆₇	—	2	12	no	—
15	C₂×#95×C₆₇	C₂²	1	6	yes	—
16	C₂×#95×C₆₇	C₂²	1	12	yes	—
17	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
18	C₂²×Q₈×C₆₇	—	2	192	no	—
19	C₂²×Q₈×C₆₇	—	2	32	no	—
20	C₂²×Q₈×C₆₇	—	2	24	no	—
21	C₂²×Q₈×C₆₇	—	2	24	no	—
22	C₂²×Q₈×C₆₇	D₄	1	16	yes	—
23	C₂³×C₄×C₆₇	—	2	12	no	—
24	C₂³×C₄×C₆₇	—	2	24	no	—
25	C₂³×C₄×C₆₇	D₄	1	12	yes	—
26	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
27	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
28	C₂³×C₄×C₆₇	C₂³	1	6	yes	—
29	C₂³×C₄×C₆₇	C₂³	1	6	yes	—
30	C₂×C₄⋊C₄×C₆₇	—	2	12	no	—
31	C₂×C₄⋊C₄×C₆₇	—	2	24	no	—
32	C₂×C₂²⋊C₄×C₆₇	—	2	6	no	—
33	C₂×C₂²⋊C₄×C₆₇	—	2	6	no	—
34	C₂×C₂²⋊C₄×C₆₇	—	2	6	no	—
35	C₂×C₂²⋊C₄×C₆₇	Q₈	1	12	yes	complemented
36	#95×C₆₇	—	2	12	no	—
37	#95×C₆₇	—	2	24	no	—
38	C₂⁴×C₆₇	—	2	4	no	—
39	C₂⁴×C₆₇	—	2	4	no	—
40	C₂⁴×C₆₇	C₂×D₄	1	6	yes	—
41	C₂⁴×C₆₇	C₂×Q₈	1	6	yes	—
42	C₂⁴×C₆₇	#41	1	2	yes	—
43	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
44	C₂×Q₈×C₆₇	—	2	256	no	—
45	C₂×Q₈×C₆₇	—	2	32	no	—
46	C₂×Q₈×C₆₇	—	2	96	no	—
47	C₂×Q₈×C₆₇	—	2	96	no	—
48	C₂²×C₄×C₆₇	—	2	96	no	—
49	C₂²×C₄×C₆₇	—	2	24	no	—
50	C₂²×C₄×C₆₇	—	2	12	no	—
51	C₂²×C₄×C₆₇	—	2	12	no	—
52	C₂²×C₄×C₆₇	—	2	48	no	—
53	C₂²×C₄×C₆₇	—	2	6	no	—
54	C₂²×C₄×C₆₇	—	2	12	no	—
55	C₂²×C₄×C₆₇	—	2	6	no	—
56	C₂²×C₄×C₆₇	—	2	12	no	—
57	C₂²×C₄×C₆₇	—	2	3	no	—
58	C₂²×C₄×C₆₇	—	2	3	no	—
59	C₂²×C₄×C₆₇	—	2	4	no	—
60	C₂²×C₄×C₆₇	—	2	4	no	—
61	C₂²×C₄×C₆₇	C₂×D₄	1	24	yes	—
62	C₂²×C₄×C₆₇	C₂×D₄	1	12	yes	—
63	C₂²×C₄×C₆₇	C₂×Q₈	1	12	yes	complemented
64	C₄⋊C₄×C₆₇	—	4	24	no	—
65	C₂²⋊C₄×C₆₇	—	2	24	no	—
66	C₂²⋊C₄×C₆₇	—	4	4	no	—
67	C₂³×C₆₇	—	2	12	no	—
68	C₂³×C₆₇	—	2	12	no	—
69	C₂³×C₆₇	—	2	8	no	—
70	C₂³×C₆₇	—	2	24	no	—
71	C₂³×C₆₇	#120	1	4	yes	—
72	C₂³×C₆₇	C₂²⋊Q₈	1	12	yes	—
73	C₂³×C₆₇	C₄⋊Q₈	1	12	yes	—
74	C₂³×C₆₇	C₂²×D₄	1	3	yes	—
75	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
76	C₂³×C₆₇	C₂²×Q₈	1	3	yes	—
77	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
78	C₂³×C₆₇	2₋¹⁺⁴	1	4	yes	—
79	Q₈×C₆₇	—	2	128	no	—
80	C₂×C₄×C₆₇	—	2	96	no	—
81	C₂×C₄×C₆₇	—	2	12	no	—
82	C₂×C₄×C₆₇	—	2	48	no	—
83	C₂×C₄×C₆₇	—	2	12	no	—
84	C₂×C₄×C₆₇	—	2	12	no	—
85	C₂×C₄×C₆₇	—	2	24	no	—
86	C₂×C₄×C₆₇	—	4	6	no	—
87	C₂×C₄×C₆₇	—	4	6	no	—
88	C₂×C₄×C₆₇	—	4	12	no	—
89	C₂×C₄×C₆₇	—	4	1	no	—
90	C₂×C₄×C₆₇	—	4	1	no	—
91	C₂×C₄×C₆₇	—	4	6	no	—
92	C₂×C₄×C₆₇	—	4	6	no	—
93	C₂×C₄×C₆₇	#120	1	24	yes	—
94	C₂²×C₆₇	—	2	4	no	—
95	C₂²×C₆₇	—	2	4	no	—
96	C₂²×C₆₇	—	2	24	no	—
97	C₂²×C₆₇	—	2	24	no	—
98	C₂²×C₆₇	#395	1	8	yes	complemented
99	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
100	C₂²×C₆₇	C₂×#120	1	6	yes	—
101	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	3	yes	—
102	C₂²×C₆₇	C₂×C₄⋊Q₈	1	3	yes	—
103	C₂²×C₆₇	#536	1	6	yes	—
104	C₂²×C₆₇	#548	1	4	yes	—
105	C₂²×C₆₇	D₄×Q₈	1	12	yes	—
106	C₄×C₆₇	—	2	24	no	—
107	C₄×C₆₇	—	4	12	no	—
108	C₄×C₆₇	—	8	1	no	—
109	C₄×C₆₇	—	8	1	no	—
110	#14874	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂×C₆₇	—	2	8	no	—
112	C₂×C₆₇	C₂×#395	1	1	yes	characteristic
113	C₂×C₆₇	#2409	1	6	yes	—
114	C₂×C₆₇	#2434	1	6	yes	—
115	C₂×C₆₇	#2614	1	1	yes	characteristic
116	C₂×C₆₇	#2846	1	1	yes	characteristic
117	C₂⁴×Q₈	C₂×C₆₇	1	1	yes	characteristic
118	C₂×#395	C₂×C₆₇	1	4	yes	complemented
119	C₂×#385	C₂×C₆₇	1	3	yes	—
120	#1446	C₂×C₆₇	1	4	yes	complemented
121	C₂²⋊#95	C₂×C₆₇	1	3	yes	—
122	C₆₇	#14874	1	1	yes	characteristic, direct product factor, Sylow
123	C₂³×Q₈	—	2	16	no	—
124	C₂³×Q₈	—	2	12	no	—
125	C₂³×Q₈	C₂²×C₆₇	1	4	yes	—
126	C₂⁴×C₄	C₂²×C₆₇	1	3	yes	—
127	C₂²×C₄⋊C₄	C₂²×C₆₇	1	6	yes	—
128	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
129	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	3	yes	—
130	#395	—	2	16	no	—
131	#385	—	2	12	no	—
132	C₂×#95	C₂²×C₆₇	1	12	yes	—
133	C₂×#95	C₂²×C₆₇	1	6	yes	—
134	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
135	C₂²×Q₈	—	2	192	no	—
136	C₂²×Q₈	—	2	32	no	—
137	C₂²×Q₈	—	2	24	no	—
138	C₂²×Q₈	—	2	24	no	—
139	C₂²×Q₈	D₄×C₆₇	1	16	yes	—
140	C₂³×C₄	—	2	12	no	—
141	C₂³×C₄	—	2	24	no	—
142	C₂³×C₄	D₄×C₆₇	1	12	yes	—
143	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
144	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
145	C₂³×C₄	C₂³×C₆₇	1	6	yes	—
146	C₂³×C₄	C₂³×C₆₇	1	6	yes	—
147	C₂×C₄⋊C₄	—	2	12	no	—
148	C₂×C₄⋊C₄	—	2	24	no	—
149	C₂×C₂²⋊C₄	—	2	6	no	—
150	C₂×C₂²⋊C₄	—	2	6	no	—
151	C₂×C₂²⋊C₄	—	2	6	no	—
152	C₂×C₂²⋊C₄	Q₈×C₆₇	1	12	yes	complemented
153	#95	—	2	12	no	—
154	#95	—	2	24	no	—
155	C₂⁴	—	2	4	no	—
156	C₂⁴	—	2	4	no	—
157	C₂⁴	C₂×D₄×C₆₇	1	6	yes	—
158	C₂⁴	C₂×Q₈×C₆₇	1	6	yes	—
159	C₂⁴	#41×C₆₇	1	2	yes	—
160	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
161	C₂×Q₈	—	2	256	no	—
162	C₂×Q₈	—	2	32	no	—
163	C₂×Q₈	—	2	96	no	—
164	C₂×Q₈	—	2	96	no	—
165	C₂²×C₄	—	2	96	no	—
166	C₂²×C₄	—	2	24	no	—
167	C₂²×C₄	—	2	12	no	—
168	C₂²×C₄	—	2	12	no	—
169	C₂²×C₄	—	2	48	no	—
170	C₂²×C₄	—	2	12	no	—
171	C₂²×C₄	—	2	6	no	—
172	C₂²×C₄	—	2	6	no	—
173	C₂²×C₄	—	2	12	no	—
174	C₂²×C₄	—	2	3	no	—
175	C₂²×C₄	—	2	3	no	—
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	C₂×D₄×C₆₇	1	24	yes	—
179	C₂²×C₄	C₂×D₄×C₆₇	1	12	yes	—
180	C₂²×C₄	C₂×Q₈×C₆₇	1	12	yes	complemented
181	C₄⋊C₄	—	4	24	no	—
182	C₂²⋊C₄	—	2	24	no	—
183	C₂²⋊C₄	—	4	4	no	—
184	C₂³	—	2	12	no	—
185	C₂³	—	2	12	no	—
186	C₂³	—	2	8	no	—
187	C₂³	—	2	24	no	—
188	C₂³	#120×C₆₇	1	4	yes	—
189	C₂³	C₂²⋊Q₈×C₆₇	1	12	yes	—
190	C₂³	C₄⋊Q₈×C₆₇	1	12	yes	—
191	C₂³	C₂²×D₄×C₆₇	1	3	yes	—
192	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
193	C₂³	C₂²×Q₈×C₆₇	1	3	yes	—
194	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
195	C₂³	2₋¹⁺⁴×C₆₇	1	4	yes	—
196	Q₈	—	2	128	no	—
197	C₂×C₄	—	2	96	no	—
198	C₂×C₄	—	2	12	no	—
199	C₂×C₄	—	2	48	no	—
200	C₂×C₄	—	2	12	no	—
201	C₂×C₄	—	2	12	no	—
202	C₂×C₄	—	2	24	no	—
203	C₂×C₄	—	4	6	no	—
204	C₂×C₄	—	4	6	no	—
205	C₂×C₄	—	4	12	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	—	4	6	no	—
209	C₂×C₄	—	4	6	no	—
210	C₂×C₄	#120×C₆₇	1	24	yes	—
211	C₂²	—	2	4	no	—
212	C₂²	—	2	4	no	—
213	C₂²	—	2	24	no	—
214	C₂²	—	2	24	no	—
215	C₂²	#395×C₆₇	1	8	yes	complemented
216	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
217	C₂²	C₂×#120×C₆₇	1	6	yes	—
218	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	3	yes	—
219	C₂²	C₂×C₄⋊Q₈×C₆₇	1	3	yes	—
220	C₂²	#536×C₆₇	1	6	yes	—
221	C₂²	#548×C₆₇	1	4	yes	—
222	C₂²	D₄×Q₈×C₆₇	1	12	yes	—
223	C₄	—	2	24	no	—
224	C₄	—	4	12	no	—
225	C₄	—	8	1	no	—
226	C₄	—	8	1	no	—
227	C₂	—	2	8	no	—
228	C₂	C₂×#395×C₆₇	1	1	yes	characteristic
229	C₂	C₆₇×#2409	1	6	yes	—
230	C₂	C₆₇×#2434	1	6	yes	—
231	C₂	C₆₇×#2614	1	1	yes	characteristic
232	C₂	C₆₇×#2846	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₆₇
prime radical	C₂⁵×C₆₇	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#395×C₆₇	C₂	semidirect
4. C₆₇×#1446	C₂	semidirect
35. C₂×C₂²⋊C₄×C₆₇	Q₈	semidirect
63. C₂²×C₄×C₆₇	C₂×Q₈	semidirect
98. C₂²×C₆₇	#395	semidirect
110. #14874	C₆₇	direct (N × H)
118. C₂×#395	C₂×C₆₇	semidirect
120. #1446	C₂×C₆₇	semidirect
122. C₆₇	#14874	direct (N × H)
152. C₂×C₂²⋊C₄	Q₈×C₆₇	semidirect
180. C₂²×C₄	C₂×Q₈×C₆₇	semidirect
215. C₂²	#395×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂⁴×Q₈×C₆₇	2
C₂×#395×C₆₇	2
C₂×#385×C₆₇	2
C₆₇×#1446	2
C₆₇×C₂²⋊#95	2
#14874	67

Subgroups — C67×#14874 #30694125

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#14874 #30694125