Subgroups — C₆₇×#22686 #30701937

Order 17152.

858 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#444×C₆₇	C₂	1	2	yes	maximal
2	C₄×#135×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2906	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1946	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1866	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1848	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1806	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1770	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×#131×C₆₇	C₂²	1	4	yes	—
11	C₄×#41×C₆₇	C₂²	1	2	yes	—
12	#444×C₆₇	—	2	8	no	—
13	C₄×Q₁₆×C₆₇	C₄	1	1	yes	characteristic, complemented
14	C₄×Q₁₆×C₆₇	C₄	1	1	yes	characteristic, complemented
15	C₄×SD₁₆×C₆₇	C₄	1	2	yes	complemented
16	C₄×SD₁₆×C₆₇	C₄	1	2	yes	complemented
17	C₄×D₈×C₆₇	C₄	1	1	yes	characteristic, complemented
18	C₄×D₈×C₆₇	C₄	1	1	yes	characteristic, complemented
19	C₂×#108×C₆₇	C₂²	1	2	yes	—
20	C₂×#108×C₆₇	C₂²	1	2	yes	—
21	#427×C₆₇	C₂²	1	1	yes	characteristic
22	#424×C₆₇	—	2	4	no	—
23	#424×C₆₇	C₂²	1	4	yes	—
24	#423×C₆₇	—	2	2	no	—
25	#423×C₆₇	C₂²	1	2	yes	—
26	C₂×C₄⋊C₈×C₆₇	—	2	2	no	—
27	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×#104×C₆₇	C₂²	1	4	yes	—
29	#416×C₆₇	C₂²	1	2	yes	—
30	C₂×#97×C₆₇	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
33	#339×C₆₇	C₂²	1	4	yes	—
34	#336×C₆₇	C₂²	1	2	yes	—
35	C₂×#41×C₆₇	C₂³	1	2	yes	—
36	#135×C₆₇	D₄	1	4	yes	—
37	#135×C₆₇	Q₈	1	4	yes	—
38	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
40	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic
41	#131×C₆₇	—	2	8	no	—
42	#131×C₆₇	—	4	4	no	—
43	C₂×M₁₆×C₆₇	—	2	4	no	—
44	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
45	C₂²×C₈×C₆₇	—	2	4	no	—
46	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
48	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
49	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
50	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
51	C₄×D₄×C₆₇	—	2	2	no	—
52	C₄×D₄×C₆₇	—	2	2	no	—
53	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
54	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
55	#117×C₆₇	—	2	2	no	—
56	#117×C₆₇	C₂³	1	2	yes	—
57	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
58	C₂×C₄²×C₆₇	—	2	2	no	—
59	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
60	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
61	#108×C₆₇	—	2	4	no	—
62	#108×C₆₇	—	4	2	no	—
63	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
64	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
65	#106×C₆₇	C₂×C₄	1	2	yes	—
66	C₄⋊C₈×C₆₇	—	2	4	no	—
67	C₄⋊C₈×C₆₇	—	2	4	no	—
68	C₄⋊C₈×C₆₇	—	2	2	no	—
69	C₄⋊C₈×C₆₇	—	2	2	no	—
70	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
71	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
72	#104×C₆₇	—	2	8	no	—
73	#103×C₆₇	C₂×C₄	1	2	yes	—
74	#103×C₆₇	C₂×C₄	1	2	yes	—
75	#102×C₆₇	C₂×C₄	1	2	yes	—
76	#102×C₆₇	C₂×C₄	1	2	yes	—
77	#97×C₆₇	—	2	4	no	—
78	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
79	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
80	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
81	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
82	#95×C₆₇	—	2	1	no	—
83	#41×C₆₇	—	2	2	no	—
84	#41×C₆₇	—	2	2	no	—
85	#41×C₆₇	C₂×D₄	1	4	yes	—
86	#41×C₆₇	C₂×Q₈	1	4	yes	—
87	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
88	C₂×D₄×C₆₇	—	2	2	no	—
89	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₆₇	—	2	2	no	—
91	C₂²×C₄×C₆₇	—	2	2	no	—
92	C₂²×C₄×C₆₇	—	2	2	no	—
93	C₂²×C₄×C₆₇	—	2	2	no	—
94	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
95	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
96	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
97	Q₁₆×C₆₇	C₄⋊C₄	1	4	yes	complemented
98	SD₁₆×C₆₇	C₄⋊C₄	1	8	yes	complemented
99	D₈×C₆₇	C₄⋊C₄	1	4	yes	complemented
100	M₁₆×C₆₇	—	2	4	no	—
101	M₁₆×C₆₇	—	2	4	no	—
102	M₁₆×C₆₇	—	4	8	no	—
103	C₂×C₈×C₆₇	—	2	2	no	—
104	C₂×C₈×C₆₇	—	2	1	no	—
105	C₂×C₈×C₆₇	—	2	2	no	—
106	C₂×C₈×C₆₇	—	2	1	no	—
107	C₂×C₈×C₆₇	—	2	4	no	—
108	C₂×C₈×C₆₇	—	4	8	no	—
109	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
114	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
115	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
116	C₄⋊C₄×C₆₇	—	2	2	no	—
117	C₄⋊C₄×C₆₇	—	2	2	no	—
118	C₄⋊C₄×C₆₇	—	4	2	no	—
119	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
120	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
121	C₂²⋊C₄×C₆₇	—	2	2	no	—
122	C₂²⋊C₄×C₆₇	—	2	2	no	—
123	C₂²⋊C₄×C₆₇	—	2	2	no	—
124	C₄²×C₆₇	—	2	2	no	—
125	C₄²×C₆₇	—	2	2	no	—
126	C₄²×C₆₇	—	2	2	no	—
127	C₄²×C₆₇	—	2	2	no	—
128	C₄²×C₆₇	—	2	2	no	—
129	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
130	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
131	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
132	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
133	C₂³×C₆₇	—	2	2	no	—
134	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
135	Q₈×C₆₇	—	2	2	no	—
136	Q₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
137	D₄×C₆₇	—	2	4	no	—
138	D₄×C₆₇	—	2	4	no	—
139	D₄×C₆₇	—	2	2	no	—
140	D₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
141	C₂×C₄×C₆₇	—	2	2	no	—
142	C₂×C₄×C₆₇	—	2	4	no	—
143	C₂×C₄×C₆₇	—	2	4	no	—
144	C₂×C₄×C₆₇	—	2	2	no	—
145	C₂×C₄×C₆₇	—	2	2	no	—
146	C₂×C₄×C₆₇	—	2	2	no	—
147	C₂×C₄×C₆₇	—	4	1	no	—
148	C₂×C₄×C₆₇	—	4	4	no	—
149	C₂×C₄×C₆₇	—	4	2	no	—
150	C₂×C₄×C₆₇	—	4	2	no	—
151	C₂×C₄×C₆₇	—	4	2	no	—
152	C₂×C₄×C₆₇	—	8	2	no	—
153	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
154	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
155	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
156	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
157	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
158	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
161	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
162	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
163	C₈×C₆₇	—	2	1	no	—
164	C₈×C₆₇	—	2	1	no	—
165	C₈×C₆₇	—	2	4	no	—
166	C₈×C₆₇	—	4	4	no	—
167	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
168	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
169	C₂²×C₆₇	—	2	1	no	—
170	C₂²×C₆₇	—	2	1	no	—
171	C₂²×C₆₇	—	2	4	no	—
172	C₂²×C₆₇	—	4	2	no	—
173	C₂²×C₆₇	—	4	2	no	—
174	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
175	C₂²×C₆₇	#547	1	1	yes	characteristic
176	C₂²×C₆₇	#554	1	1	yes	characteristic
177	C₄×C₆₇	—	2	4	no	—
178	C₄×C₆₇	—	2	1	no	—
179	C₄×C₆₇	—	2	1	no	—
180	C₄×C₆₇	—	2	1	no	—
181	C₄×C₆₇	—	4	2	no	—
182	C₄×C₆₇	—	4	2	no	—
183	C₄×C₆₇	—	8	2	no	—
184	C₄×C₆₇	#444	1	2	yes	complemented
185	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
186	C₄×C₆₇	#520	1	1	yes	characteristic
187	C₄×C₆₇	D₄²	1	1	yes	characteristic
188	C₄×C₆₇	D₄×Q₈	1	1	yes	characteristic
189	#22686	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
190	C₂×C₆₇	—	2	1	no	—
191	C₂×C₆₇	—	2	1	no	—
192	C₂×C₆₇	—	4	4	no	—
193	C₂×C₆₇	D₄×C₄⋊C₄	1	1	yes	characteristic
194	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
195	C₂×C₆₇	#2957	1	1	yes	characteristic
196	C₂×#444	C₂×C₆₇	1	2	yes	—
197	C₄×#135	C₂×C₆₇	1	1	yes	characteristic
198	#2906	C₂×C₆₇	1	2	yes	complemented
199	#1946	C₂×C₆₇	1	2	yes	complemented
200	#1866	C₂×C₆₇	1	4	yes	complemented
201	#1848	C₂×C₆₇	1	1	yes	characteristic, complemented
202	#1806	C₂×C₆₇	1	2	yes	complemented
203	#1770	C₂×C₆₇	1	1	yes	characteristic, complemented
204	C₆₇	#22686	1	1	yes	characteristic, direct product factor, Sylow
205	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
206	C₂×#131	C₂²×C₆₇	1	4	yes	—
207	C₄×#41	C₂²×C₆₇	1	2	yes	—
208	#444	—	2	8	no	—
209	C₄×Q₁₆	C₄×C₆₇	1	1	yes	characteristic, complemented
210	C₄×Q₁₆	C₄×C₆₇	1	1	yes	characteristic, complemented
211	C₄×SD₁₆	C₄×C₆₇	1	2	yes	complemented
212	C₄×SD₁₆	C₄×C₆₇	1	2	yes	complemented
213	C₄×D₈	C₄×C₆₇	1	1	yes	characteristic, complemented
214	C₄×D₈	C₄×C₆₇	1	1	yes	characteristic, complemented
215	C₂×#108	C₂²×C₆₇	1	2	yes	—
216	C₂×#108	C₂²×C₆₇	1	2	yes	—
217	#427	C₂²×C₆₇	1	1	yes	characteristic
218	#424	—	2	4	no	—
219	#424	C₂²×C₆₇	1	4	yes	—
220	#423	—	2	2	no	—
221	#423	C₂²×C₆₇	1	2	yes	—
222	C₂×C₄⋊C₈	—	2	2	no	—
223	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
224	C₂×#104	C₂²×C₆₇	1	4	yes	—
225	#416	C₂²×C₆₇	1	2	yes	—
226	C₂×#97	C₂²×C₆₇	1	2	yes	—
227	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
228	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
229	#339	C₂²×C₆₇	1	4	yes	—
230	#336	C₂²×C₆₇	1	2	yes	—
231	C₂×#41	C₂³×C₆₇	1	2	yes	—
232	#135	D₄×C₆₇	1	4	yes	—
233	#135	Q₈×C₆₇	1	4	yes	—
234	C₂×Q₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
235	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	—
236	C₂×D₈	C₂×C₄×C₆₇	1	1	yes	characteristic
237	#131	—	2	8	no	—
238	#131	—	4	4	no	—
239	C₂×M₁₆	—	2	4	no	—
240	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
241	C₂²×C₈	—	2	4	no	—
242	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
243	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
244	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
245	C₄×Q₈	C₂×C₄×C₆₇	1	2	yes	—
246	C₄×Q₈	C₂×C₄×C₆₇	1	2	yes	—
247	C₄×D₄	—	2	2	no	—
248	C₄×D₄	—	2	2	no	—
249	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
250	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
251	#117	—	2	2	no	—
252	#117	C₂³×C₆₇	1	2	yes	—
253	C₂×C₄⋊C₄	—	2	1	no	—
254	C₂×C₄²	—	2	2	no	—
255	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
256	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
257	#108	—	2	4	no	—
258	#108	—	4	2	no	—
259	#107	C₂×C₄×C₆₇	1	1	yes	characteristic
260	#107	C₂×C₄×C₆₇	1	1	yes	characteristic
261	#106	C₂×C₄×C₆₇	1	2	yes	—
262	C₄⋊C₈	—	2	4	no	—
263	C₄⋊C₈	—	2	4	no	—
264	C₄⋊C₈	—	2	2	no	—
265	C₄⋊C₈	—	2	2	no	—
266	C₄⋊C₈	D₄×C₆₇	1	2	yes	—
267	C₄⋊C₈	D₄×C₆₇	1	2	yes	—
268	#104	—	2	8	no	—
269	#103	C₂×C₄×C₆₇	1	2	yes	—
270	#103	C₂×C₄×C₆₇	1	2	yes	—
271	#102	C₂×C₄×C₆₇	1	2	yes	—
272	#102	C₂×C₄×C₆₇	1	2	yes	—
273	#97	—	2	4	no	—
274	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
275	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
276	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
277	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
278	#95	—	2	1	no	—
279	#41	—	2	2	no	—
280	#41	—	2	2	no	—
281	#41	C₂×D₄×C₆₇	1	4	yes	—
282	#41	C₂×Q₈×C₆₇	1	4	yes	—
283	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
284	C₂×D₄	—	2	2	no	—
285	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
286	C₂²×C₄	—	2	2	no	—
287	C₂²×C₄	—	2	2	no	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	2	2	no	—
290	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
291	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
292	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
293	Q₁₆	C₄⋊C₄×C₆₇	1	4	yes	complemented
294	SD₁₆	C₄⋊C₄×C₆₇	1	8	yes	complemented
295	D₈	C₄⋊C₄×C₆₇	1	4	yes	complemented
296	M₁₆	—	2	4	no	—
297	M₁₆	—	2	4	no	—
298	M₁₆	—	4	8	no	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	2	1	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	2	1	no	—
303	C₂×C₈	—	2	4	no	—
304	C₂×C₈	—	4	8	no	—
305	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
306	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
307	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
308	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
309	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
310	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
311	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
312	C₄⋊C₄	—	2	2	no	—
313	C₄⋊C₄	—	2	2	no	—
314	C₄⋊C₄	—	4	2	no	—
315	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
316	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
317	C₂²⋊C₄	—	2	2	no	—
318	C₂²⋊C₄	—	2	2	no	—
319	C₂²⋊C₄	—	2	2	no	—
320	C₄²	—	2	2	no	—
321	C₄²	—	2	2	no	—
322	C₄²	—	2	2	no	—
323	C₄²	—	2	2	no	—
324	C₄²	—	2	2	no	—
325	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
326	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
327	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
328	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
329	C₂³	—	2	2	no	—
330	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
331	Q₈	—	2	2	no	—
332	Q₈	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
333	D₄	—	2	4	no	—
334	D₄	—	2	4	no	—
335	D₄	—	2	2	no	—
336	D₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	4	no	—
339	C₂×C₄	—	2	4	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	4	4	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	8	2	no	—
349	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
350	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
351	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
352	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
353	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
354	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
355	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
356	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
357	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
358	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
359	C₈	—	2	1	no	—
360	C₈	—	2	1	no	—
361	C₈	—	2	4	no	—
362	C₈	—	4	4	no	—
363	C₈	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
364	C₈	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	2	4	no	—
368	C₂²	—	4	2	no	—
369	C₂²	—	4	2	no	—
370	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
371	C₂²	#547×C₆₇	1	1	yes	characteristic
372	C₂²	#554×C₆₇	1	1	yes	characteristic
373	C₄	—	2	4	no	—
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	4	2	no	—
378	C₄	—	4	2	no	—
379	C₄	—	8	2	no	—
380	C₄	#444×C₆₇	1	2	yes	complemented
381	C₄	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
382	C₄	#520×C₆₇	1	1	yes	characteristic
383	C₄	D₄²×C₆₇	1	1	yes	characteristic
384	C₄	D₄×Q₈×C₆₇	1	1	yes	characteristic
385	C₂	—	2	1	no	—
386	C₂	—	2	1	no	—
387	C₂	—	4	4	no	—
388	C₂	D₄×C₄⋊C₄×C₆₇	1	1	yes	characteristic
389	C₂	C₂×#444×C₆₇	1	1	yes	characteristic
390	C₂	C₆₇×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#135×C₆₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₄⋊C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#2906	C₂	semidirect
4. C₆₇×#1946	C₂	semidirect
5. C₆₇×#1866	C₂	semidirect
6. C₆₇×#1848	C₂	semidirect
7. C₆₇×#1806	C₂	semidirect
8. C₆₇×#1770	C₂	semidirect
13. C₄×Q₁₆×C₆₇	C₄	semidirect
14. C₄×Q₁₆×C₆₇	C₄	semidirect
15. C₄×SD₁₆×C₆₇	C₄	semidirect
16. C₄×SD₁₆×C₆₇	C₄	semidirect
17. C₄×D₈×C₆₇	C₄	semidirect
18. C₄×D₈×C₆₇	C₄	semidirect
97. Q₁₆×C₆₇	C₄⋊C₄	semidirect
98. SD₁₆×C₆₇	C₄⋊C₄	semidirect
99. D₈×C₆₇	C₄⋊C₄	semidirect
184. C₄×C₆₇	#444	semidirect
189. #22686	C₆₇	direct (N × H)
198. #2906	C₂×C₆₇	semidirect
199. #1946	C₂×C₆₇	semidirect
200. #1866	C₂×C₆₇	semidirect
201. #1848	C₂×C₆₇	semidirect
202. #1806	C₂×C₆₇	semidirect
203. #1770	C₂×C₆₇	semidirect
204. C₆₇	#22686	direct (N × H)
209. C₄×Q₁₆	C₄×C₆₇	semidirect
210. C₄×Q₁₆	C₄×C₆₇	semidirect
211. C₄×SD₁₆	C₄×C₆₇	semidirect
212. C₄×SD₁₆	C₄×C₆₇	semidirect
213. C₄×D₈	C₄×C₆₇	semidirect
214. C₄×D₈	C₄×C₆₇	semidirect
293. Q₁₆	C₄⋊C₄×C₆₇	semidirect
294. SD₁₆	C₄⋊C₄×C₆₇	semidirect
295. D₈	C₄⋊C₄×C₆₇	semidirect
380. C₄	#444×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#444×C₆₇	2
C₄×#135×C₆₇	2
C₆₇×#2906	2
C₆₇×#1946	2
C₆₇×#1866	2
C₆₇×#1848	2
C₆₇×#1806	2
C₆₇×#1770	2
#22686	67

Subgroups — C67×#22686 #30701937

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22686 #30701937