Subgroups — C₂²×#362×C₆₇ #30712641

Order 17152.

1134 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×#108×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#362×C₆₇	C₂	1	12	yes	maximal, direct product factor
5	C₂³×Q₈×C₆₇	—	2	1	no	—
6	C₂²×Q₁₆×C₆₇	—	2	3	no	—
7	C₂²×Q₁₆×C₆₇	—	2	1	no	—
8	C₂²×Q₁₆×C₆₇	—	2	3	no	—
9	C₂²×Q₁₆×C₆₇	C₄	1	6	yes	—
10	C₂²×Q₁₆×C₆₇	C₄	1	2	yes	—
11	C₂²×Q₁₆×C₆₇	C₂²	1	6	yes	—
12	C₂²×M₁₆×C₆₇	—	2	1	no	—
13	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×M₃₂×C₆₇	—	2	3	no	—
15	C₂×M₃₂×C₆₇	C₂²	1	6	yes	—
16	C₂²×C₁₆×C₆₇	—	2	1	no	—
17	C₂×#108×C₆₇	—	2	3	no	—
18	C₂×#108×C₆₇	C₂²	1	6	yes	—
19	#362×C₆₇	C₂²	1	16	yes	direct product factor
20	C₂²×Q₈×C₆₇	—	2	6	no	—
21	C₂²×Q₈×C₆₇	—	2	2	no	—
22	C₂²×Q₈×C₆₇	—	2	6	no	—
23	C₂²×Q₈×C₆₇	—	4	3	no	—
24	C₂²×Q₈×C₆₇	—	4	1	no	—
25	C₂²×Q₈×C₆₇	—	4	3	no	—
26	C₂³×C₄×C₆₇	—	4	1	no	—
27	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
28	C₂×Q₁₆×C₆₇	—	2	6	no	—
29	C₂×Q₁₆×C₆₇	—	2	12	no	—
30	C₂×Q₁₆×C₆₇	—	2	6	no	—
31	C₂×Q₁₆×C₆₇	—	2	6	no	—
32	C₂×Q₁₆×C₆₇	—	2	6	no	—
33	C₂×Q₁₆×C₆₇	—	2	6	no	—
34	C₂×Q₁₆×C₆₇	C₂×C₄	1	12	yes	—
35	C₂×Q₁₆×C₆₇	C₂×C₄	1	12	yes	—
36	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
37	C₂×M₁₆×C₆₇	—	2	6	no	—
38	C₂×M₁₆×C₆₇	—	4	3	no	—
39	C₂²×C₈×C₆₇	—	4	1	no	—
40	C₂²×C₈×C₆₇	C₂×C₄	1	3	yes	—
41	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₆₇	D₄	1	3	yes	—
43	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₆₇	D₄	1	3	yes	—
45	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
46	M₃₂×C₆₇	—	2	6	no	—
47	M₃₂×C₆₇	C₂³	1	4	yes	—
48	C₂×C₁₆×C₆₇	—	2	6	no	—
49	#108×C₆₇	—	2	6	no	—
50	#108×C₆₇	C₂³	1	4	yes	—
51	C₂⁴×C₆₇	SD₁₆	1	1	yes	characteristic
52	C₂×Q₈×C₆₇	—	2	12	no	—
53	C₂×Q₈×C₆₇	—	2	12	no	—
54	C₂×Q₈×C₆₇	—	2	4	no	—
55	C₂×Q₈×C₆₇	—	4	12	no	—
56	C₂×Q₈×C₆₇	—	4	6	no	—
57	C₂×Q₈×C₆₇	—	4	6	no	—
58	C₂×Q₈×C₆₇	—	4	6	no	—
59	C₂×Q₈×C₆₇	—	4	6	no	—
60	C₂×Q₈×C₆₇	—	4	6	no	—
61	C₂²×C₄×C₆₇	—	4	6	no	—
62	C₂²×C₄×C₆₇	—	4	2	no	—
63	C₂²×C₄×C₆₇	—	4	6	no	—
64	C₂²×C₄×C₆₇	C₂²⋊C₄	1	3	yes	—
65	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	D₈	1	3	yes	—
67	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	SD₁₆	1	3	yes	—
69	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
70	Q₁₆×C₆₇	—	2	12	no	—
71	Q₁₆×C₆₇	—	2	12	no	—
72	Q₁₆×C₆₇	—	2	4	no	—
73	Q₁₆×C₆₇	C₂²×C₄	1	8	yes	—
74	M₁₆×C₆₇	—	2	4	no	—
75	M₁₆×C₆₇	—	4	6	no	—
76	C₂×C₈×C₆₇	—	4	6	no	—
77	C₂×C₈×C₆₇	C₂²⋊C₄	1	3	yes	—
78	C₂×C₈×C₆₇	C₂²⋊C₄	1	6	yes	—
79	C₂×C₈×C₆₇	C₂²⋊C₄	1	3	yes	—
80	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
81	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
82	C₂×C₈×C₆₇	C₂×D₄	1	3	yes	—
83	C₂×C₈×C₆₇	C₂×D₄	1	3	yes	—
84	C₂×C₈×C₆₇	C₂×D₄	1	3	yes	—
85	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
86	C₁₆×C₆₇	—	2	4	no	—
87	C₂³×C₆₇	—	2	4	no	—
88	C₂³×C₆₇	#102	1	3	yes	—
89	C₂³×C₆₇	#102	1	1	yes	characteristic
90	C₂³×C₆₇	C₂×SD₁₆	1	3	yes	—
91	Q₈×C₆₇	—	2	8	no	—
92	Q₈×C₆₇	—	4	4	no	—
93	Q₈×C₆₇	—	4	12	no	—
94	Q₈×C₆₇	—	4	12	no	—
95	C₂×C₄×C₆₇	—	4	4	no	—
96	C₂×C₄×C₆₇	—	4	12	no	—
97	C₂×C₄×C₆₇	—	4	12	no	—
98	C₂×C₄×C₆₇	#102	1	6	yes	—
99	C₂×C₄×C₆₇	#102	1	3	yes	—
100	C₂×C₄×C₆₇	#102	1	3	yes	—
101	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	3	yes	—
102	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	3	yes	—
103	C₂×C₄×C₆₇	C₂×D₈	1	3	yes	—
104	C₂×C₄×C₆₇	C₂×D₈	1	3	yes	—
105	C₂×C₄×C₆₇	C₂×SD₁₆	1	3	yes	—
106	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
107	C₈×C₆₇	—	4	4	no	—
108	C₈×C₆₇	C₂×C₂²⋊C₄	1	3	yes	—
109	C₈×C₆₇	C₂×C₂²⋊C₄	1	3	yes	—
110	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
111	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
112	C₂²×C₆₇	—	2	6	no	—
113	C₂²×C₆₇	—	2	6	no	—
114	C₂²×C₆₇	#362	1	4	yes	direct product factor
115	C₂²×C₆₇	C₂×#102	1	3	yes	—
116	C₂²×C₆₇	C₂×#102	1	3	yes	—
117	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
118	C₄×C₆₇	—	4	8	no	—
119	C₄×C₆₇	C₂×#102	1	3	yes	—
120	C₄×C₆₇	C₂×#102	1	3	yes	—
121	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
122	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
123	C₂²×#362	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂×C₆₇	—	2	1	no	—
125	C₂×C₆₇	—	2	3	no	—
126	C₂×C₆₇	C₂×#362	1	6	yes	direct product factor
127	C₂×C₆₇	C₂²×#102	1	1	yes	characteristic
128	C₂³×Q₁₆	C₂×C₆₇	1	1	yes	characteristic
129	C₂²×M₃₂	C₂×C₆₇	1	1	yes	characteristic
130	C₂²×#108	C₂×C₆₇	1	1	yes	characteristic
131	C₂×#362	C₂×C₆₇	1	12	yes	direct product factor
132	C₆₇	C₂²×#362	1	1	yes	characteristic, direct product factor, Sylow
133	C₂³×Q₈	—	2	1	no	—
134	C₂²×Q₁₆	—	2	3	no	—
135	C₂²×Q₁₆	—	2	1	no	—
136	C₂²×Q₁₆	—	2	3	no	—
137	C₂²×Q₁₆	C₄×C₆₇	1	6	yes	—
138	C₂²×Q₁₆	C₄×C₆₇	1	2	yes	—
139	C₂²×Q₁₆	C₂²×C₆₇	1	6	yes	—
140	C₂²×M₁₆	—	2	1	no	—
141	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
142	C₂×M₃₂	—	2	3	no	—
143	C₂×M₃₂	C₂²×C₆₇	1	6	yes	—
144	C₂²×C₁₆	—	2	1	no	—
145	C₂×#108	—	2	3	no	—
146	C₂×#108	C₂²×C₆₇	1	6	yes	—
147	#362	C₂²×C₆₇	1	16	yes	direct product factor
148	C₂²×Q₈	—	2	6	no	—
149	C₂²×Q₈	—	2	2	no	—
150	C₂²×Q₈	—	2	6	no	—
151	C₂²×Q₈	—	4	1	no	—
152	C₂²×Q₈	—	4	3	no	—
153	C₂²×Q₈	—	4	3	no	—
154	C₂³×C₄	—	4	1	no	—
155	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
156	C₂×Q₁₆	—	2	6	no	—
157	C₂×Q₁₆	—	2	12	no	—
158	C₂×Q₁₆	—	2	6	no	—
159	C₂×Q₁₆	—	2	6	no	—
160	C₂×Q₁₆	—	2	6	no	—
161	C₂×Q₁₆	—	2	6	no	—
162	C₂×Q₁₆	C₂×C₄×C₆₇	1	12	yes	—
163	C₂×Q₁₆	C₂×C₄×C₆₇	1	12	yes	—
164	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
165	C₂×M₁₆	—	2	6	no	—
166	C₂×M₁₆	—	4	3	no	—
167	C₂²×C₈	—	4	1	no	—
168	C₂²×C₈	C₂×C₄×C₆₇	1	3	yes	—
169	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
170	C₂²×C₈	D₄×C₆₇	1	3	yes	—
171	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
172	C₂²×C₈	D₄×C₆₇	1	3	yes	—
173	C₂²×C₈	C₂³×C₆₇	1	3	yes	—
174	M₃₂	—	2	6	no	—
175	M₃₂	C₂³×C₆₇	1	4	yes	—
176	C₂×C₁₆	—	2	6	no	—
177	#108	—	2	6	no	—
178	#108	C₂³×C₆₇	1	4	yes	—
179	C₂⁴	SD₁₆×C₆₇	1	1	yes	characteristic
180	C₂×Q₈	—	2	12	no	—
181	C₂×Q₈	—	2	12	no	—
182	C₂×Q₈	—	2	4	no	—
183	C₂×Q₈	—	4	12	no	—
184	C₂×Q₈	—	4	6	no	—
185	C₂×Q₈	—	4	6	no	—
186	C₂×Q₈	—	4	6	no	—
187	C₂×Q₈	—	4	6	no	—
188	C₂×Q₈	—	4	6	no	—
189	C₂²×C₄	—	4	6	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	—	4	6	no	—
192	C₂²×C₄	C₂²⋊C₄×C₆₇	1	3	yes	—
193	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
194	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
195	C₂²×C₄	D₈×C₆₇	1	3	yes	—
196	C₂²×C₄	SD₁₆×C₆₇	1	3	yes	—
197	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
198	Q₁₆	—	2	12	no	—
199	Q₁₆	—	2	12	no	—
200	Q₁₆	—	2	4	no	—
201	Q₁₆	C₂²×C₄×C₆₇	1	8	yes	—
202	M₁₆	—	2	4	no	—
203	M₁₆	—	4	6	no	—
204	C₂×C₈	—	4	6	no	—
205	C₂×C₈	C₂²⋊C₄×C₆₇	1	3	yes	—
206	C₂×C₈	C₂²⋊C₄×C₆₇	1	6	yes	—
207	C₂×C₈	C₂²⋊C₄×C₆₇	1	3	yes	—
208	C₂×C₈	C₂²×C₄×C₆₇	1	3	yes	—
209	C₂×C₈	C₂²×C₄×C₆₇	1	3	yes	—
210	C₂×C₈	C₂×D₄×C₆₇	1	3	yes	—
211	C₂×C₈	C₂×D₄×C₆₇	1	3	yes	—
212	C₂×C₈	C₂×D₄×C₆₇	1	3	yes	—
213	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
214	C₁₆	—	2	4	no	—
215	C₂³	—	2	4	no	—
216	C₂³	#102×C₆₇	1	3	yes	—
217	C₂³	#102×C₆₇	1	1	yes	characteristic
218	C₂³	C₂×SD₁₆×C₆₇	1	3	yes	—
219	Q₈	—	2	8	no	—
220	Q₈	—	4	4	no	—
221	Q₈	—	4	12	no	—
222	Q₈	—	4	12	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	—	4	12	no	—
225	C₂×C₄	—	4	12	no	—
226	C₂×C₄	#102×C₆₇	1	6	yes	—
227	C₂×C₄	#102×C₆₇	1	3	yes	—
228	C₂×C₄	#102×C₆₇	1	3	yes	—
229	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	3	yes	—
230	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	3	yes	—
231	C₂×C₄	C₂×D₈×C₆₇	1	3	yes	—
232	C₂×C₄	C₂×D₈×C₆₇	1	3	yes	—
233	C₂×C₄	C₂×SD₁₆×C₆₇	1	3	yes	—
234	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
235	C₈	—	4	4	no	—
236	C₈	C₂×C₂²⋊C₄×C₆₇	1	3	yes	—
237	C₈	C₂×C₂²⋊C₄×C₆₇	1	3	yes	—
238	C₈	C₂³×C₄×C₆₇	1	1	yes	characteristic
239	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
240	C₂²	—	2	6	no	—
241	C₂²	—	2	6	no	—
242	C₂²	#362×C₆₇	1	4	yes	direct product factor
243	C₂²	C₂×#102×C₆₇	1	3	yes	—
244	C₂²	C₂×#102×C₆₇	1	3	yes	—
245	C₂²	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
246	C₄	—	4	8	no	—
247	C₄	C₂×#102×C₆₇	1	3	yes	—
248	C₄	C₂×#102×C₆₇	1	3	yes	—
249	C₄	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
250	C₄	C₂²×D₈×C₆₇	1	1	yes	characteristic
251	C₂	—	2	1	no	—
252	C₂	—	2	3	no	—
253	C₂	C₂×#362×C₆₇	1	6	yes	direct product factor
254	C₂	C₂²×#102×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	#102
commutator	C₈	C₂³×C₄×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	C₂⁴×C₆₇	SD₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ C₂³×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×#362×C₆₇	C₂	direct (N × H)
19. #362×C₆₇	C₂²	direct (N × H)
114. C₂²×C₆₇	#362	direct (N × H)
123. C₂²×#362	C₆₇	direct (N × H)
126. C₂×C₆₇	C₂×#362	direct (N × H)
131. C₂×#362	C₂×C₆₇	direct (N × H)
132. C₆₇	C₂²×#362	direct (N × H)
147. #362	C₂²×C₆₇	direct (N × H)
242. C₂²	#362×C₆₇	direct (N × H)
253. C₂	C₂×#362×C₆₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Q₁₆×C₆₇	2
C₂²×M₃₂×C₆₇	2
C₂²×#108×C₆₇	2
C₂×#362×C₆₇	2
C₂²×#362	67

Subgroups — C22×#362×C67 #30712641

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#362×C₆₇ #30712641