Subgroups — C₂²⋊C₄×Q₁₆×C₆₇ #30701701

Order 17152.

1398 conjugacy classes of subgroups.

The subgroup lattice has 364 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	C₂×C₄×Q₁₆×C₆₇	C₂	1	2	yes	maximal, complemented
3	Q₈×C₂²⋊C₄×C₆₇	C₂	1	2	yes	maximal
4	C₆₇×#1969	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1878	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1808	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#1787	C₂	1	2	yes	maximal
8	C₈×C₂²⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂³×Q₈×C₆₇	C₂²	1	2	yes	—
10	C₂²×Q₁₆×C₆₇	—	2	8	no	—
11	C₂²×Q₁₆×C₆₇	C₄	1	8	yes	complemented
12	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
13	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×Q₈×C₆₇	C₂²	1	4	yes	—
17	C₄×Q₁₆×C₆₇	—	2	8	no	—
18	C₂×#107×C₆₇	C₂²	1	2	yes	—
19	C₂×#103×C₆₇	C₂²	1	4	yes	—
20	C₂×#103×C₆₇	C₂²	1	4	yes	—
21	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
23	#391×C₆₇	—	2	4	no	—
24	#391×C₆₇	C₂²	1	4	yes	—
25	#380×C₆₇	—	2	2	no	—
26	#380×C₆₇	C₂²	1	2	yes	—
27	C₄×C₂²⋊C₄×C₆₇	—	2	2	no	—
28	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
29	#340×C₆₇	C₂²	1	2	yes	—
30	#336×C₆₇	C₂²	1	2	yes	—
31	C₂²×Q₈×C₆₇	—	2	2	no	—
32	C₂²×Q₈×C₆₇	—	2	4	no	—
33	C₂²×Q₈×C₆₇	—	2	8	no	—
34	C₂²×Q₈×C₆₇	—	4	4	no	—
35	C₂²×Q₈×C₆₇	C₂×C₄	1	8	yes	—
36	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
37	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
38	C₂³×C₄×C₆₇	—	2	2	no	—
39	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
40	C₂×Q₁₆×C₆₇	—	2	8	no	—
41	C₂×Q₁₆×C₆₇	—	2	32	no	—
42	C₂×Q₁₆×C₆₇	—	2	4	no	—
43	C₂×Q₁₆×C₆₇	C₂×C₄	1	8	yes	—
44	C₂×Q₁₆×C₆₇	C₂×C₄	1	4	yes	—
45	C₂×Q₁₆×C₆₇	C₂×C₄	1	4	yes	—
46	C₂×Q₁₆×C₆₇	D₄	1	8	yes	—
47	C₂²×C₈×C₆₇	—	2	4	no	—
48	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
49	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
50	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₄×Q₈×C₆₇	—	2	8	no	—
53	C₄×Q₈×C₆₇	—	4	4	no	—
54	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
55	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
56	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
57	C₂×C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
58	C₂×C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
59	C₂×C₄²×C₆₇	—	2	4	no	—
60	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
61	#107×C₆₇	—	2	4	no	—
62	#103×C₆₇	—	2	8	no	—
63	#103×C₆₇	—	4	4	no	—
64	C₂²⋊C₈×C₆₇	—	2	2	no	—
65	C₄×C₈×C₆₇	—	2	4	no	—
66	#95×C₆₇	—	2	4	no	—
67	#95×C₆₇	—	2	4	no	—
68	#95×C₆₇	—	2	2	no	—
69	#95×C₆₇	D₄	1	2	yes	—
70	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
71	C₂×Q₈×C₆₇	—	2	8	no	—
72	C₂×Q₈×C₆₇	—	2	2	no	—
73	C₂×Q₈×C₆₇	—	2	2	no	—
74	C₂×Q₈×C₆₇	—	2	4	no	—
75	C₂×Q₈×C₆₇	—	2	16	no	—
76	C₂×Q₈×C₆₇	—	2	4	no	—
77	C₂×Q₈×C₆₇	—	2	4	no	—
78	C₂×Q₈×C₆₇	—	4	8	no	—
79	C₂×Q₈×C₆₇	—	4	8	no	—
80	C₂×Q₈×C₆₇	—	4	16	no	—
81	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
82	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
83	C₂×Q₈×C₆₇	C₂×D₄	1	8	yes	—
84	C₂²×C₄×C₆₇	—	2	2	no	—
85	C₂²×C₄×C₆₇	—	2	2	no	—
86	C₂²×C₄×C₆₇	—	2	2	no	—
87	C₂²×C₄×C₆₇	—	2	8	no	—
88	C₂²×C₄×C₆₇	—	2	2	no	—
89	C₂²×C₄×C₆₇	—	2	4	no	—
90	C₂²×C₄×C₆₇	—	4	8	no	—
91	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
93	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
95	C₂²×C₄×C₆₇	#41	1	2	yes	—
96	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
97	Q₁₆×C₆₇	—	2	16	no	—
98	Q₁₆×C₆₇	—	2	8	no	—
99	Q₁₆×C₆₇	C₂²⋊C₄	1	16	yes	direct product factor
100	C₂×C₈×C₆₇	—	2	2	no	—
101	C₂×C₈×C₆₇	—	2	8	no	—
102	C₂×C₈×C₆₇	—	2	2	no	—
103	C₂×C₈×C₆₇	—	2	2	no	—
104	C₂×C₈×C₆₇	—	2	2	no	—
105	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
106	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
107	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
108	C₄⋊C₄×C₆₇	—	2	4	no	—
109	C₄⋊C₄×C₆₇	—	2	4	no	—
110	C₄⋊C₄×C₆₇	—	4	8	no	—
111	C₂²⋊C₄×C₆₇	—	2	2	no	—
112	C₂²⋊C₄×C₆₇	—	4	4	no	—
113	C₂²⋊C₄×C₆₇	Q₁₆	1	4	yes	direct product factor
114	C₄²×C₆₇	—	2	2	no	—
115	C₄²×C₆₇	—	2	2	no	—
116	C₄²×C₆₇	—	4	8	no	—
117	C₂³×C₆₇	—	2	2	no	—
118	C₂³×C₆₇	—	2	2	no	—
119	C₂³×C₆₇	C₄×D₄	1	2	yes	—
120	C₂³×C₆₇	C₂×Q₁₆	1	2	yes	—
121	C₂³×C₆₇	#135	1	2	yes	—
122	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
123	Q₈×C₆₇	—	2	4	no	—
124	Q₈×C₆₇	—	2	8	no	—
125	Q₈×C₆₇	—	2	4	no	—
126	Q₈×C₆₇	—	2	4	no	—
127	Q₈×C₆₇	—	4	16	no	—
128	Q₈×C₆₇	—	4	4	no	—
129	Q₈×C₆₇	C₂×C₂²⋊C₄	1	8	yes	—
130	C₂×C₄×C₆₇	—	2	4	no	—
131	C₂×C₄×C₆₇	—	2	2	no	—
132	C₂×C₄×C₆₇	—	2	8	no	—
133	C₂×C₄×C₆₇	—	2	2	no	—
134	C₂×C₄×C₆₇	—	2	2	no	—
135	C₂×C₄×C₆₇	—	2	2	no	—
136	C₂×C₄×C₆₇	—	2	2	no	—
137	C₂×C₄×C₆₇	—	2	4	no	—
138	C₂×C₄×C₆₇	—	2	4	no	—
139	C₂×C₄×C₆₇	—	2	2	no	—
140	C₂×C₄×C₆₇	—	2	2	no	—
141	C₂×C₄×C₆₇	—	2	2	no	—
142	C₂×C₄×C₆₇	—	4	16	no	—
143	C₂×C₄×C₆₇	—	4	4	no	—
144	C₂×C₄×C₆₇	—	4	4	no	—
145	C₂×C₄×C₆₇	—	4	4	no	—
146	C₂×C₄×C₆₇	—	4	4	no	—
147	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
148	C₂×C₄×C₆₇	C₂×Q₁₆	1	4	yes	complemented
149	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
150	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
151	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
152	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
153	C₈×C₆₇	—	2	2	no	—
154	C₈×C₆₇	—	4	2	no	—
155	C₈×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
156	C₂²×C₆₇	—	2	2	no	—
157	C₂²×C₆₇	—	2	2	no	—
158	C₂²×C₆₇	—	2	4	no	—
159	C₂²×C₆₇	—	2	4	no	—
160	C₂²×C₆₇	C₄×Q₁₆	1	4	yes	complemented
161	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
162	C₂²×C₆₇	D₄²	1	2	yes	—
163	C₂²×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
164	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
165	C₂²×C₆₇	#578	1	2	yes	—
166	C₄×C₆₇	—	2	2	no	—
167	C₄×C₆₇	—	2	8	no	—
168	C₄×C₆₇	—	2	4	no	—
169	C₄×C₆₇	—	4	4	no	—
170	C₄×C₆₇	—	4	2	no	—
171	C₄×C₆₇	—	8	4	no	—
172	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
173	C₄×C₆₇	#518	1	1	yes	characteristic
174	C₄×C₆₇	#546	1	2	yes	—
175	C₂²⋊C₄×Q₁₆	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
176	C₂×C₆₇	—	2	4	no	—
177	C₂×C₆₇	D₄×C₂²⋊C₄	1	1	yes	characteristic
178	C₂×C₆₇	C₂×C₄×Q₁₆	1	1	yes	characteristic
179	C₂×C₆₇	#2950	1	1	yes	characteristic
180	C₂×C₆₇	#3285	1	2	yes	—
181	C₂×C₆₇	D₄×Q₁₆	1	2	yes	—
182	C₂³×Q₁₆	C₂×C₆₇	1	1	yes	characteristic
183	C₂×C₄×Q₁₆	C₂×C₆₇	1	2	yes	complemented
184	Q₈×C₂²⋊C₄	C₂×C₆₇	1	2	yes	—
185	#1969	C₂×C₆₇	1	2	yes	complemented
186	#1878	C₂×C₆₇	1	4	yes	complemented
187	#1808	C₂×C₆₇	1	1	yes	characteristic
188	#1787	C₂×C₆₇	1	2	yes	—
189	C₈×C₂²⋊C₄	C₂×C₆₇	1	1	yes	characteristic
190	C₆₇	C₂²⋊C₄×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
191	C₂³×Q₈	C₂²×C₆₇	1	2	yes	—
192	C₂²×Q₁₆	—	2	8	no	—
193	C₂²×Q₁₆	C₄×C₆₇	1	8	yes	complemented
194	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	—
195	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
196	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
197	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
198	C₂×C₄×Q₈	C₂²×C₆₇	1	4	yes	—
199	C₄×Q₁₆	—	2	8	no	—
200	C₂×#107	C₂²×C₆₇	1	2	yes	—
201	C₂×#103	C₂²×C₆₇	1	4	yes	—
202	C₂×#103	C₂²×C₆₇	1	4	yes	—
203	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
204	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
205	#391	—	2	4	no	—
206	#391	C₂²×C₆₇	1	4	yes	—
207	#380	—	2	2	no	—
208	#380	C₂²×C₆₇	1	2	yes	—
209	C₄×C₂²⋊C₄	—	2	2	no	—
210	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
211	#340	C₂²×C₆₇	1	2	yes	—
212	#336	C₂²×C₆₇	1	2	yes	—
213	C₂²×Q₈	—	2	2	no	—
214	C₂²×Q₈	—	2	4	no	—
215	C₂²×Q₈	—	2	8	no	—
216	C₂²×Q₈	—	4	4	no	—
217	C₂²×Q₈	C₂×C₄×C₆₇	1	8	yes	—
218	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
219	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
220	C₂³×C₄	—	2	2	no	—
221	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
222	C₂×Q₁₆	—	2	8	no	—
223	C₂×Q₁₆	—	2	32	no	—
224	C₂×Q₁₆	—	2	4	no	—
225	C₂×Q₁₆	C₂×C₄×C₆₇	1	8	yes	—
226	C₂×Q₁₆	C₂×C₄×C₆₇	1	4	yes	—
227	C₂×Q₁₆	C₂×C₄×C₆₇	1	4	yes	—
228	C₂×Q₁₆	D₄×C₆₇	1	8	yes	—
229	C₂²×C₈	—	2	4	no	—
230	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
231	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
232	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
233	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
234	C₄×Q₈	—	2	8	no	—
235	C₄×Q₈	—	4	4	no	—
236	C₂×C₄⋊C₄	—	2	4	no	—
237	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
238	C₂×C₂²⋊C₄	—	2	2	no	—
239	C₂×C₂²⋊C₄	D₄×C₆₇	1	1	yes	characteristic
240	C₂×C₂²⋊C₄	D₄×C₆₇	1	1	yes	characteristic
241	C₂×C₄²	—	2	4	no	—
242	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
243	#107	—	2	4	no	—
244	#103	—	2	8	no	—
245	#103	—	4	4	no	—
246	C₂²⋊C₈	—	2	2	no	—
247	C₄×C₈	—	2	4	no	—
248	#95	—	2	4	no	—
249	#95	—	2	4	no	—
250	#95	—	2	2	no	—
251	#95	D₄×C₆₇	1	2	yes	—
252	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
253	C₂×Q₈	—	2	8	no	—
254	C₂×Q₈	—	2	2	no	—
255	C₂×Q₈	—	2	2	no	—
256	C₂×Q₈	—	2	4	no	—
257	C₂×Q₈	—	2	16	no	—
258	C₂×Q₈	—	2	4	no	—
259	C₂×Q₈	—	2	4	no	—
260	C₂×Q₈	—	4	8	no	—
261	C₂×Q₈	—	4	8	no	—
262	C₂×Q₈	—	4	16	no	—
263	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
264	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
265	C₂×Q₈	C₂×D₄×C₆₇	1	8	yes	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	2	8	no	—
270	C₂²×C₄	—	2	2	no	—
271	C₂²×C₄	—	2	4	no	—
272	C₂²×C₄	—	4	8	no	—
273	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
274	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
275	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
276	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
277	C₂²×C₄	#41×C₆₇	1	2	yes	—
278	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
279	Q₁₆	—	2	16	no	—
280	Q₁₆	—	2	8	no	—
281	Q₁₆	C₂²⋊C₄×C₆₇	1	16	yes	direct product factor
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	8	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
288	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
289	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
290	C₄⋊C₄	—	2	4	no	—
291	C₄⋊C₄	—	2	4	no	—
292	C₄⋊C₄	—	4	8	no	—
293	C₂²⋊C₄	—	2	2	no	—
294	C₂²⋊C₄	—	4	4	no	—
295	C₂²⋊C₄	Q₁₆×C₆₇	1	4	yes	direct product factor
296	C₄²	—	2	2	no	—
297	C₄²	—	2	2	no	—
298	C₄²	—	4	8	no	—
299	C₂³	—	2	2	no	—
300	C₂³	—	2	2	no	—
301	C₂³	C₄×D₄×C₆₇	1	2	yes	—
302	C₂³	C₂×Q₁₆×C₆₇	1	2	yes	—
303	C₂³	#135×C₆₇	1	2	yes	—
304	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
305	Q₈	—	2	4	no	—
306	Q₈	—	2	8	no	—
307	Q₈	—	2	4	no	—
308	Q₈	—	2	4	no	—
309	Q₈	—	4	16	no	—
310	Q₈	—	4	4	no	—
311	Q₈	C₂×C₂²⋊C₄×C₆₇	1	8	yes	—
312	C₂×C₄	—	2	4	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	8	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	4	no	—
320	C₂×C₄	—	2	4	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	4	16	no	—
325	C₂×C₄	—	4	4	no	—
326	C₂×C₄	—	4	4	no	—
327	C₂×C₄	—	4	4	no	—
328	C₂×C₄	—	4	4	no	—
329	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
330	C₂×C₄	C₂×Q₁₆×C₆₇	1	4	yes	complemented
331	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
332	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
333	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
334	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
335	C₈	—	2	2	no	—
336	C₈	—	4	2	no	—
337	C₈	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
338	C₂²	—	2	4	no	—
339	C₂²	—	2	4	no	—
340	C₂²	—	2	2	no	—
341	C₂²	—	2	2	no	—
342	C₂²	C₄×Q₁₆×C₆₇	1	4	yes	complemented
343	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
344	C₂²	D₄²×C₆₇	1	2	yes	—
345	C₂²	C₂²×Q₁₆×C₆₇	1	1	yes	characteristic
346	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
347	C₂²	#578×C₆₇	1	2	yes	—
348	C₄	—	2	4	no	—
349	C₄	—	2	2	no	—
350	C₄	—	2	8	no	—
351	C₄	—	4	2	no	—
352	C₄	—	4	4	no	—
353	C₄	—	8	4	no	—
354	C₄	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
355	C₄	#518×C₆₇	1	1	yes	characteristic
356	C₄	#546×C₆₇	1	2	yes	—
357	C₂	—	2	4	no	—
358	C₂	D₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
359	C₂	C₂×C₄×Q₁₆×C₆₇	1	1	yes	characteristic
360	C₂	C₆₇×#2950	1	1	yes	characteristic
361	C₂	C₆₇×#3285	1	2	yes	—
362	C₂	D₄×Q₁₆×C₆₇	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Q₁₆×C₆₇	C₂	semidirect
4. C₆₇×#1969	C₂	semidirect
5. C₆₇×#1878	C₂	semidirect
11. C₂²×Q₁₆×C₆₇	C₄	semidirect
99. Q₁₆×C₆₇	C₂²⋊C₄	direct (N × H)
113. C₂²⋊C₄×C₆₇	Q₁₆	direct (N × H)
148. C₂×C₄×C₆₇	C₂×Q₁₆	semidirect
160. C₂²×C₆₇	C₄×Q₁₆	semidirect
175. C₂²⋊C₄×Q₁₆	C₆₇	direct (N × H)
183. C₂×C₄×Q₁₆	C₂×C₆₇	semidirect
185. #1969	C₂×C₆₇	semidirect
186. #1878	C₂×C₆₇	semidirect
190. C₆₇	C₂²⋊C₄×Q₁₆	direct (N × H)
193. C₂²×Q₁₆	C₄×C₆₇	semidirect
281. Q₁₆	C₂²⋊C₄×C₆₇	direct (N × H)
295. C₂²⋊C₄	Q₁₆×C₆₇	direct (N × H)
330. C₂×C₄	C₂×Q₁₆×C₆₇	semidirect
342. C₂²	C₄×Q₁₆×C₆₇	semidirect

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
C₂×C₄×Q₁₆×C₆₇	2
Q₈×C₂²⋊C₄×C₆₇	2
C₆₇×#1969	2
C₆₇×#1878	2
C₆₇×#1808	2
C₆₇×#1787	2
C₈×C₂²⋊C₄×C₆₇	2
C₂²⋊C₄×Q₁₆	67

Subgroups — C22⋊C4×Q16×C67 #30701701

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²⋊C₄×Q₁₆×C₆₇ #30701701