Subgroups — C₆₇×#33237 #30712488

Order 17152.

946 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#488×C₆₇	C₂	1	4	yes	maximal, complemented
2	C₆₇×#2859	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2858	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1961	C₂	1	2	yes	maximal
5	C₆₇×#1960	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1959	C₂	1	4	yes	maximal, complemented
7	C₄×#97×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
9	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×#125×C₆₇	C₂²	1	2	yes	—
11	C₂×#124×C₆₇	C₂²	1	2	yes	—
12	C₂×#124×C₆₇	C₂²	1	2	yes	—
13	#488×C₆₇	—	2	16	no	—
14	C₂×#103×C₆₇	C₂²	1	8	yes	—
15	C₂×#102×C₆₇	C₂²	1	8	yes	—
16	C₂×#97×C₆₇	C₂²	1	2	yes	—
17	C₂×#97×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
19	#390×C₆₇	—	2	4	no	—
20	#389×C₆₇	—	2	4	no	—
21	#384×C₆₇	C₂²	1	2	yes	—
22	#382×C₆₇	C₂²	1	2	yes	—
23	C₄³×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
25	C₂²×D₄×C₆₇	—	2	2	no	—
26	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
27	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
28	C₄⋊Q₈×C₆₇	—	2	4	no	—
29	#127×C₆₇	—	2	4	no	—
30	#125×C₆₇	—	2	8	no	—
31	#124×C₆₇	—	2	8	no	—
32	#124×C₆₇	—	2	8	no	—
33	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
34	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
35	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
36	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
37	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
38	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
39	C₂×C₄²×C₆₇	D₄	1	4	yes	—
40	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
43	#103×C₆₇	—	2	16	no	—
44	#102×C₆₇	—	2	16	no	—
45	#97×C₆₇	—	2	4	no	—
46	#97×C₆₇	—	2	4	no	—
47	C₄×C₈×C₆₇	D₄	1	8	yes	complemented
48	#95×C₆₇	—	2	4	no	—
49	C₂⁴×C₆₇	—	2	2	no	—
50	C₂×Q₈×C₆₇	—	2	4	no	—
51	C₂×Q₈×C₆₇	—	2	4	no	—
52	C₂×Q₈×C₆₇	—	2	4	no	—
53	C₂×D₄×C₆₇	—	2	4	no	—
54	C₂×D₄×C₆₇	—	2	4	no	—
55	C₂×D₄×C₆₇	—	2	4	no	—
56	C₂×D₄×C₆₇	—	4	8	no	—
57	C₂×D₄×C₆₇	—	4	4	no	—
58	C₂²×C₄×C₆₇	—	2	2	no	—
59	C₂²×C₄×C₆₇	—	2	4	no	—
60	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
64	C₂×C₈×C₆₇	—	2	4	no	—
65	C₂×C₈×C₆₇	—	2	4	no	—
66	C₂×C₈×C₆₇	C₂×D₄	1	8	yes	—
67	C₄⋊C₄×C₆₇	—	2	8	no	—
68	C₄⋊C₄×C₆₇	—	4	8	no	—
69	C₄⋊C₄×C₆₇	—	4	4	no	—
70	C₄⋊C₄×C₆₇	—	4	8	no	—
71	C₂²⋊C₄×C₆₇	—	4	8	no	—
72	C₂²⋊C₄×C₆₇	—	4	8	no	—
73	C₄²×C₆₇	—	4	4	no	—
74	C₄²×C₆₇	C₂×D₄	1	4	yes	—
75	C₄²×C₆₇	#41	1	4	yes	—
76	C₄²×C₆₇	#41	1	4	yes	—
77	C₂³×C₆₇	—	4	2	no	—
78	C₂³×C₆₇	—	4	2	no	—
79	C₂³×C₆₇	—	4	2	no	—
80	C₂³×C₆₇	—	4	8	no	—
81	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
82	Q₈×C₆₇	—	4	8	no	—
83	D₄×C₆₇	—	4	8	no	—
84	D₄×C₆₇	—	8	8	no	—
85	C₂×C₄×C₆₇	—	2	4	no	—
86	C₂×C₄×C₆₇	—	2	4	no	—
87	C₂×C₄×C₆₇	—	2	4	no	—
88	C₂×C₄×C₆₇	—	4	2	no	—
89	C₂×C₄×C₆₇	—	4	2	no	—
90	C₂×C₄×C₆₇	—	4	2	no	—
91	C₂×C₄×C₆₇	—	4	4	no	—
92	C₂×C₄×C₆₇	—	4	8	no	—
93	C₂×C₄×C₆₇	#124	1	4	yes	—
94	C₂×C₄×C₆₇	#124	1	4	yes	—
95	C₂×C₄×C₆₇	#127	1	4	yes	—
96	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
97	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
98	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
99	C₈×C₆₇	—	2	8	no	—
100	C₂²×C₆₇	—	8	2	no	—
101	C₂²×C₆₇	—	8	4	no	—
102	C₂²×C₆₇	—	8	4	no	—
103	C₂²×C₆₇	—	8	4	no	—
104	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
105	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
106	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
107	C₂²×C₆₇	#575	1	4	yes	—
108	C₄×C₆₇	—	2	4	no	—
109	C₄×C₆₇	—	2	4	no	—
110	C₄×C₆₇	—	2	4	no	—
111	C₄×C₆₇	—	8	2	no	—
112	C₄×C₆₇	—	8	4	no	—
113	C₄×C₆₇	#532	1	4	yes	—
114	#33237	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂×C₆₇	—	16	2	no	—
116	C₂×C₆₇	#2858	1	1	yes	characteristic
117	C₂×C₆₇	#3140	1	2	yes	—
118	C₂×C₆₇	#3142	1	2	yes	—
119	C₂×C₆₇	#3150	1	2	yes	—
120	C₂×#488	C₂×C₆₇	1	4	yes	complemented
121	#2859	C₂×C₆₇	1	1	yes	characteristic, complemented
122	#2858	C₂×C₆₇	1	1	yes	characteristic
123	#1961	C₂×C₆₇	1	2	yes	—
124	#1960	C₂×C₆₇	1	2	yes	complemented
125	#1959	C₂×C₆₇	1	4	yes	complemented
126	C₄×#97	C₂×C₆₇	1	1	yes	characteristic, complemented
127	C₆₇	#33237	1	1	yes	characteristic, direct product factor, Sylow
128	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
129	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
130	C₂×#125	C₂²×C₆₇	1	2	yes	—
131	C₂×#124	C₂²×C₆₇	1	2	yes	—
132	C₂×#124	C₂²×C₆₇	1	2	yes	—
133	#488	—	2	16	no	—
134	C₂×#103	C₂²×C₆₇	1	8	yes	—
135	C₂×#102	C₂²×C₆₇	1	8	yes	—
136	C₂×#97	C₂²×C₆₇	1	2	yes	—
137	C₂×#97	C₂²×C₆₇	1	2	yes	—
138	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
139	#390	—	2	4	no	—
140	#389	—	2	4	no	—
141	#384	C₂²×C₆₇	1	2	yes	—
142	#382	C₂²×C₆₇	1	2	yes	—
143	C₄³	C₂²×C₆₇	1	1	yes	characteristic
144	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
145	C₂²×D₄	—	2	2	no	—
146	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
147	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
148	C₄⋊Q₈	—	2	4	no	—
149	#127	—	2	4	no	—
150	#125	—	2	8	no	—
151	#124	—	2	8	no	—
152	#124	—	2	8	no	—
153	C₂×C₄⋊C₄	—	2	4	no	—
154	C₂×C₄⋊C₄	—	2	2	no	—
155	C₂×C₄⋊C₄	—	2	4	no	—
156	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
157	C₂×C₂²⋊C₄	—	2	4	no	—
158	C₂×C₂²⋊C₄	—	2	4	no	—
159	C₂×C₄²	D₄×C₆₇	1	4	yes	—
160	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
161	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
162	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
163	#103	—	2	16	no	—
164	#102	—	2	16	no	—
165	#97	—	2	4	no	—
166	#97	—	2	4	no	—
167	C₄×C₈	D₄×C₆₇	1	8	yes	complemented
168	#95	—	2	4	no	—
169	C₂⁴	—	2	2	no	—
170	C₂×Q₈	—	2	4	no	—
171	C₂×Q₈	—	2	4	no	—
172	C₂×Q₈	—	2	4	no	—
173	C₂×D₄	—	2	4	no	—
174	C₂×D₄	—	2	4	no	—
175	C₂×D₄	—	2	4	no	—
176	C₂×D₄	—	4	8	no	—
177	C₂×D₄	—	4	4	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
181	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
182	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
183	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	C₂×D₄×C₆₇	1	8	yes	—
187	C₄⋊C₄	—	2	8	no	—
188	C₄⋊C₄	—	4	8	no	—
189	C₄⋊C₄	—	4	4	no	—
190	C₄⋊C₄	—	4	8	no	—
191	C₂²⋊C₄	—	4	8	no	—
192	C₂²⋊C₄	—	4	8	no	—
193	C₄²	—	4	4	no	—
194	C₄²	C₂×D₄×C₆₇	1	4	yes	—
195	C₄²	#41×C₆₇	1	4	yes	—
196	C₄²	#41×C₆₇	1	4	yes	—
197	C₂³	—	4	2	no	—
198	C₂³	—	4	2	no	—
199	C₂³	—	4	2	no	—
200	C₂³	—	4	8	no	—
201	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
202	Q₈	—	4	8	no	—
203	D₄	—	4	8	no	—
204	D₄	—	8	8	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	4	8	no	—
213	C₂×C₄	#124×C₆₇	1	4	yes	—
214	C₂×C₄	#124×C₆₇	1	4	yes	—
215	C₂×C₄	#127×C₆₇	1	4	yes	—
216	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
217	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
218	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
219	C₈	—	2	8	no	—
220	C₂²	—	8	2	no	—
221	C₂²	—	8	4	no	—
222	C₂²	—	8	4	no	—
223	C₂²	—	8	4	no	—
224	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
225	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
226	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
227	C₂²	#575×C₆₇	1	4	yes	—
228	C₄	—	2	4	no	—
229	C₄	—	2	4	no	—
230	C₄	—	2	4	no	—
231	C₄	—	8	2	no	—
232	C₄	—	8	4	no	—
233	C₄	#532×C₆₇	1	4	yes	—
234	C₂	—	16	2	no	—
235	C₂	C₆₇×#2858	1	1	yes	characteristic
236	C₂	C₆₇×#3140	1	2	yes	—
237	C₂	C₆₇×#3142	1	2	yes	—
238	C₂	C₆₇×#3150	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#127×C₆₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#488×C₆₇	C₂	semidirect
2. C₆₇×#2859	C₂	semidirect
5. C₆₇×#1960	C₂	semidirect
6. C₆₇×#1959	C₂	semidirect
7. C₄×#97×C₆₇	C₂	semidirect
47. C₄×C₈×C₆₇	D₄	semidirect
114. #33237	C₆₇	direct (N × H)
120. C₂×#488	C₂×C₆₇	semidirect
121. #2859	C₂×C₆₇	semidirect
124. #1960	C₂×C₆₇	semidirect
125. #1959	C₂×C₆₇	semidirect
126. C₄×#97	C₂×C₆₇	semidirect
127. C₆₇	#33237	direct (N × H)
167. C₄×C₈	D₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#488×C₆₇	2
C₆₇×#2859	2
C₆₇×#2858	2
C₆₇×#1961	2
C₆₇×#1960	2
C₆₇×#1959	2
C₄×#97×C₆₇	2
#33237	67

Subgroups — C67×#33237 #30712488

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33237 #30712488