Subgroups — C₆₇×#21548 #30700799

Order 17152.

914 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3054	C₂	1	1	yes	maximal, characteristic
2	C₂×#443×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2929	C₂	1	2	yes	maximal, complemented
4	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1975	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1946	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1938	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1897	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1811	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1772	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1764	C₂	1	2	yes	maximal, complemented
12	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#131×C₆₇	C₂²	1	2	yes	—
14	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	#525×C₆₇	C₂²	1	2	yes	—
17	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
18	#470×C₆₇	C₄	1	2	yes	complemented
19	#469×C₆₇	C₄	1	2	yes	complemented
20	#468×C₆₇	C₄	1	4	yes	complemented
21	#443×C₆₇	—	2	4	no	—
22	#435×C₆₇	—	2	4	no	—
23	C₈×D₄×C₆₇	—	2	4	no	—
24	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
26	#427×C₆₇	C₂²	1	1	yes	characteristic
27	#424×C₆₇	C₂²	1	2	yes	—
28	C₂×#104×C₆₇	C₂²	1	2	yes	—
29	C₂×#104×C₆₇	C₂²	1	2	yes	—
30	#416×C₆₇	C₂²	1	2	yes	—
31	#408×C₆₇	C₂²	1	2	yes	—
32	#407×C₆₇	C₂²	1	2	yes	—
33	#405×C₆₇	—	2	4	no	—
34	#405×C₆₇	C₂²	1	2	yes	—
35	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
39	#341×C₆₇	C₂²	1	2	yes	—
40	#339×C₆₇	C₂²	1	2	yes	—
41	#339×C₆₇	C₂²	1	2	yes	—
42	C₂×#41×C₆₇	C₂³	1	2	yes	—
43	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
44	#135×C₆₇	—	2	2	no	—
45	#135×C₆₇	—	2	2	no	—
46	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
48	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic
49	#131×C₆₇	—	2	4	no	—
50	#131×C₆₇	—	2	4	no	—
51	C₂×M₁₆×C₆₇	—	2	2	no	—
52	C₂×M₁₆×C₆₇	—	2	2	no	—
53	C₂×M₁₆×C₆₇	—	2	2	no	—
54	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
55	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
56	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
57	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
58	C₂²×C₈×C₆₇	—	2	2	no	—
59	C₂²×C₈×C₆₇	—	2	2	no	—
60	C₂²×C₈×C₆₇	—	2	2	no	—
61	C₂²×C₈×C₆₇	D₄	1	2	yes	—
62	C₂²×C₈×C₆₇	D₄	1	2	yes	—
63	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
67	#123×C₆₇	—	2	2	no	—
68	C₂²⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
69	#121×C₆₇	C₂×C₄	1	4	yes	—
70	#120×C₆₇	—	2	2	no	—
71	C₄×D₄×C₆₇	—	2	4	no	—
72	#117×C₆₇	—	2	2	no	—
73	#117×C₆₇	—	2	2	no	—
74	#117×C₆₇	C₂³	1	2	yes	—
75	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
76	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
77	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
79	#108×C₆₇	—	2	2	no	—
80	#108×C₆₇	—	2	2	no	—
81	#107×C₆₇	C₂×C₄	1	2	yes	—
82	#106×C₆₇	C₂×C₄	1	2	yes	—
83	C₄⋊C₈×C₆₇	—	2	4	no	—
84	#104×C₆₇	—	2	4	no	—
85	#104×C₆₇	—	2	4	no	—
86	#103×C₆₇	C₂×C₄	1	4	yes	—
87	#102×C₆₇	C₂×C₄	1	4	yes	—
88	C₂²⋊C₈×C₆₇	—	2	4	no	—
89	C₂²⋊C₈×C₆₇	—	2	4	no	—
90	#97×C₆₇	—	2	2	no	—
91	#97×C₆₇	—	2	2	no	—
92	#97×C₆₇	—	2	2	no	—
93	C₄×C₈×C₆₇	—	2	2	no	—
94	C₄×C₈×C₆₇	—	2	2	no	—
95	C₄×C₈×C₆₇	D₄	1	4	yes	complemented
96	C₂⁴×C₆₇	#41	1	1	yes	characteristic
97	#41×C₆₇	—	2	2	no	—
98	#41×C₆₇	—	2	2	no	—
99	#41×C₆₇	—	2	2	no	—
100	#41×C₆₇	—	2	2	no	—
101	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
102	C₂×D₄×C₆₇	—	2	4	no	—
103	C₂×D₄×C₆₇	—	2	2	no	—
104	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
105	C₂²×C₄×C₆₇	—	2	1	no	—
106	C₂²×C₄×C₆₇	—	2	1	no	—
107	C₂²×C₄×C₆₇	—	2	1	no	—
108	C₂²×C₄×C₆₇	—	2	1	no	—
109	C₂²×C₄×C₆₇	—	2	1	no	—
110	C₂²×C₄×C₆₇	—	2	1	no	—
111	C₂²×C₄×C₆₇	—	2	2	no	—
112	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
113	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
114	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
115	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
116	Q₁₆×C₆₇	—	2	2	no	—
117	SD₁₆×C₆₇	—	2	4	no	—
118	D₈×C₆₇	—	2	2	no	—
119	M₁₆×C₆₇	—	2	4	no	—
120	M₁₆×C₆₇	—	2	4	no	—
121	M₁₆×C₆₇	—	2	2	no	—
122	M₁₆×C₆₇	—	2	2	no	—
123	M₁₆×C₆₇	—	2	2	no	—
124	M₁₆×C₆₇	—	2	2	no	—
125	M₁₆×C₆₇	—	4	4	no	—
126	C₂×C₈×C₆₇	—	2	1	no	—
127	C₂×C₈×C₆₇	—	2	1	no	—
128	C₂×C₈×C₆₇	—	2	1	no	—
129	C₂×C₈×C₆₇	—	2	1	no	—
130	C₂×C₈×C₆₇	—	2	4	no	—
131	C₂×C₈×C₆₇	—	2	4	no	—
132	C₂×C₈×C₆₇	—	2	2	no	—
133	C₂×C₈×C₆₇	—	2	2	no	—
134	C₂×C₈×C₆₇	—	4	4	no	—
135	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
136	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
137	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
138	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
139	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
140	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
141	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
142	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
143	C₂×C₈×C₆₇	#41	1	2	yes	—
144	C₂×C₈×C₆₇	#41	1	2	yes	—
145	C₄⋊C₄×C₆₇	—	2	4	no	—
146	C₄⋊C₄×C₆₇	—	2	4	no	—
147	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
148	C₂²⋊C₄×C₆₇	—	2	4	no	—
149	C₂²⋊C₄×C₆₇	—	2	4	no	—
150	C₂²⋊C₄×C₆₇	—	2	2	no	—
151	C₂²⋊C₄×C₆₇	—	2	4	no	—
152	C₄²×C₆₇	—	2	2	no	—
153	C₄²×C₆₇	—	2	1	no	—
154	C₄²×C₆₇	—	2	1	no	—
155	C₄²×C₆₇	C₂×D₄	1	2	yes	—
156	C₄²×C₆₇	#41	1	2	yes	—
157	C₂³×C₆₇	—	2	1	no	—
158	C₂³×C₆₇	—	2	1	no	—
159	C₂³×C₆₇	—	2	2	no	—
160	C₂³×C₆₇	—	4	2	no	—
161	C₂³×C₆₇	C₄×D₄	1	2	yes	—
162	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
163	Q₈×C₆₇	—	2	2	no	—
164	Q₈×C₆₇	—	2	2	no	—
165	D₄×C₆₇	—	2	2	no	—
166	D₄×C₆₇	—	2	2	no	—
167	D₄×C₆₇	—	4	8	no	—
168	D₄×C₆₇	—	4	2	no	—
169	D₄×C₆₇	—	4	2	no	—
170	C₂×C₄×C₆₇	—	2	1	no	—
171	C₂×C₄×C₆₇	—	2	1	no	—
172	C₂×C₄×C₆₇	—	2	2	no	—
173	C₂×C₄×C₆₇	—	2	2	no	—
174	C₂×C₄×C₆₇	—	2	2	no	—
175	C₂×C₄×C₆₇	—	2	2	no	—
176	C₂×C₄×C₆₇	—	2	4	no	—
177	C₂×C₄×C₆₇	—	2	2	no	—
178	C₂×C₄×C₆₇	—	2	2	no	—
179	C₂×C₄×C₆₇	—	2	2	no	—
180	C₂×C₄×C₆₇	—	4	2	no	—
181	C₂×C₄×C₆₇	—	4	2	no	—
182	C₂×C₄×C₆₇	—	4	2	no	—
183	C₂×C₄×C₆₇	—	4	2	no	—
184	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
185	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
186	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
187	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
188	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
189	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
190	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
191	C₈×C₆₇	—	2	2	no	—
192	C₈×C₆₇	—	2	2	no	—
193	C₈×C₆₇	—	4	2	no	—
194	C₈×C₆₇	—	4	2	no	—
195	C₈×C₆₇	C₄×D₄	1	2	yes	—
196	C₈×C₆₇	C₄×D₄	1	2	yes	—
197	C₈×C₆₇	#121	1	2	yes	—
198	C₈×C₆₇	#121	1	2	yes	—
199	C₂²×C₆₇	—	2	2	no	—
200	C₂²×C₆₇	—	2	4	no	—
201	C₂²×C₆₇	—	2	2	no	—
202	C₂²×C₆₇	—	4	2	no	—
203	C₂²×C₆₇	—	4	2	no	—
204	C₂²×C₆₇	—	4	4	no	—
205	C₂²×C₆₇	—	4	2	no	—
206	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
207	C₂²×C₆₇	#525	1	1	yes	characteristic
208	C₂²×C₆₇	#529	1	1	yes	characteristic
209	C₄×C₆₇	—	2	2	no	—
210	C₄×C₆₇	—	4	2	no	—
211	C₄×C₆₇	—	4	2	no	—
212	C₄×C₆₇	—	4	4	no	—
213	C₄×C₆₇	—	4	2	no	—
214	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
215	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
216	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
217	C₄×C₆₇	#532	1	1	yes	characteristic
218	#21548	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
219	C₂×C₆₇	—	2	1	no	—
220	C₂×C₆₇	—	2	1	no	—
221	C₂×C₆₇	—	4	2	no	—
222	C₂×C₆₇	—	8	2	no	—
223	C₂×C₆₇	C₄×#121	1	1	yes	characteristic
224	C₂×C₆₇	#2955	1	2	yes	—
225	#3054	C₂×C₆₇	1	1	yes	characteristic
226	C₂×#443	C₂×C₆₇	1	1	yes	characteristic, complemented
227	#2929	C₂×C₆₇	1	2	yes	complemented
228	C₂×#405	C₂×C₆₇	1	1	yes	characteristic, complemented
229	#1975	C₂×C₆₇	1	1	yes	characteristic, complemented
230	#1946	C₂×C₆₇	1	1	yes	characteristic, complemented
231	#1938	C₂×C₆₇	1	2	yes	complemented
232	#1897	C₂×C₆₇	1	2	yes	complemented
233	#1811	C₂×C₆₇	1	1	yes	characteristic, complemented
234	#1772	C₂×C₆₇	1	1	yes	characteristic, complemented
235	#1764	C₂×C₆₇	1	2	yes	complemented
236	C₆₇	#21548	1	1	yes	characteristic, direct product factor, Sylow
237	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
238	C₂×#131	C₂²×C₆₇	1	2	yes	—
239	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
240	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
241	#525	C₂²×C₆₇	1	2	yes	—
242	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
243	#470	C₄×C₆₇	1	2	yes	complemented
244	#469	C₄×C₆₇	1	2	yes	complemented
245	#468	C₄×C₆₇	1	4	yes	complemented
246	#443	—	2	4	no	—
247	#435	—	2	4	no	—
248	C₈×D₄	—	2	4	no	—
249	C₂×#108	C₂²×C₆₇	1	1	yes	characteristic
250	C₂×#108	C₂²×C₆₇	1	1	yes	characteristic
251	#427	C₂²×C₆₇	1	1	yes	characteristic
252	#424	C₂²×C₆₇	1	2	yes	—
253	C₂×#104	C₂²×C₆₇	1	2	yes	—
254	C₂×#104	C₂²×C₆₇	1	2	yes	—
255	#416	C₂²×C₆₇	1	2	yes	—
256	#408	C₂²×C₆₇	1	2	yes	—
257	#407	C₂²×C₆₇	1	2	yes	—
258	#405	—	2	4	no	—
259	#405	C₂²×C₆₇	1	2	yes	—
260	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
261	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
262	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
263	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
264	#341	C₂²×C₆₇	1	2	yes	—
265	#339	C₂²×C₆₇	1	2	yes	—
266	#339	C₂²×C₆₇	1	2	yes	—
267	C₂×#41	C₂³×C₆₇	1	2	yes	—
268	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
269	#135	—	2	2	no	—
270	#135	—	2	2	no	—
271	C₂×Q₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
272	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	—
273	C₂×D₈	C₂×C₄×C₆₇	1	1	yes	characteristic
274	#131	—	2	4	no	—
275	#131	—	2	4	no	—
276	C₂×M₁₆	—	2	2	no	—
277	C₂×M₁₆	—	2	2	no	—
278	C₂×M₁₆	—	2	2	no	—
279	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
280	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
281	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
282	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
283	C₂²×C₈	—	2	2	no	—
284	C₂²×C₈	—	2	2	no	—
285	C₂²×C₈	—	2	2	no	—
286	C₂²×C₈	D₄×C₆₇	1	2	yes	—
287	C₂²×C₈	D₄×C₆₇	1	2	yes	—
288	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
289	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
290	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
291	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
292	#123	—	2	2	no	—
293	C₂²⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
294	#121	C₂×C₄×C₆₇	1	4	yes	—
295	#120	—	2	2	no	—
296	C₄×D₄	—	2	4	no	—
297	#117	—	2	2	no	—
298	#117	—	2	2	no	—
299	#117	C₂³×C₆₇	1	2	yes	—
300	C₂×C₄⋊C₄	—	2	1	no	—
301	C₂×C₂²⋊C₄	—	2	1	no	—
302	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
303	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
304	#108	—	2	2	no	—
305	#108	—	2	2	no	—
306	#107	C₂×C₄×C₆₇	1	2	yes	—
307	#106	C₂×C₄×C₆₇	1	2	yes	—
308	C₄⋊C₈	—	2	4	no	—
309	#104	—	2	4	no	—
310	#104	—	2	4	no	—
311	#103	C₂×C₄×C₆₇	1	4	yes	—
312	#102	C₂×C₄×C₆₇	1	4	yes	—
313	C₂²⋊C₈	—	2	4	no	—
314	C₂²⋊C₈	—	2	4	no	—
315	#97	—	2	2	no	—
316	#97	—	2	2	no	—
317	#97	—	2	2	no	—
318	C₄×C₈	—	2	2	no	—
319	C₄×C₈	—	2	2	no	—
320	C₄×C₈	D₄×C₆₇	1	4	yes	complemented
321	C₂⁴	#41×C₆₇	1	1	yes	characteristic
322	#41	—	2	2	no	—
323	#41	—	2	2	no	—
324	#41	—	2	2	no	—
325	#41	—	2	2	no	—
326	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
327	C₂×D₄	—	2	4	no	—
328	C₂×D₄	—	2	2	no	—
329	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
330	C₂²×C₄	—	2	1	no	—
331	C₂²×C₄	—	2	1	no	—
332	C₂²×C₄	—	2	1	no	—
333	C₂²×C₄	—	2	1	no	—
334	C₂²×C₄	—	2	1	no	—
335	C₂²×C₄	—	2	1	no	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
338	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
339	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
340	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
341	Q₁₆	—	2	2	no	—
342	SD₁₆	—	2	4	no	—
343	D₈	—	2	2	no	—
344	M₁₆	—	2	4	no	—
345	M₁₆	—	2	4	no	—
346	M₁₆	—	2	2	no	—
347	M₁₆	—	2	2	no	—
348	M₁₆	—	2	2	no	—
349	M₁₆	—	2	2	no	—
350	M₁₆	—	4	4	no	—
351	C₂×C₈	—	2	1	no	—
352	C₂×C₈	—	2	1	no	—
353	C₂×C₈	—	2	1	no	—
354	C₂×C₈	—	2	1	no	—
355	C₂×C₈	—	2	4	no	—
356	C₂×C₈	—	2	4	no	—
357	C₂×C₈	—	2	2	no	—
358	C₂×C₈	—	2	2	no	—
359	C₂×C₈	—	4	4	no	—
360	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
361	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
362	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
363	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
364	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
365	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
366	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
367	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
368	C₂×C₈	#41×C₆₇	1	2	yes	—
369	C₂×C₈	#41×C₆₇	1	2	yes	—
370	C₄⋊C₄	—	2	4	no	—
371	C₄⋊C₄	—	2	4	no	—
372	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
373	C₂²⋊C₄	—	2	4	no	—
374	C₂²⋊C₄	—	2	4	no	—
375	C₂²⋊C₄	—	2	2	no	—
376	C₂²⋊C₄	—	2	4	no	—
377	C₄²	—	2	1	no	—
378	C₄²	—	2	2	no	—
379	C₄²	—	2	1	no	—
380	C₄²	C₂×D₄×C₆₇	1	2	yes	—
381	C₄²	#41×C₆₇	1	2	yes	—
382	C₂³	—	2	1	no	—
383	C₂³	—	2	1	no	—
384	C₂³	—	2	2	no	—
385	C₂³	—	4	2	no	—
386	C₂³	C₄×D₄×C₆₇	1	2	yes	—
387	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
388	Q₈	—	2	2	no	—
389	Q₈	—	2	2	no	—
390	D₄	—	2	2	no	—
391	D₄	—	2	2	no	—
392	D₄	—	4	8	no	—
393	D₄	—	4	2	no	—
394	D₄	—	4	2	no	—
395	C₂×C₄	—	2	1	no	—
396	C₂×C₄	—	2	1	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	4	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	4	2	no	—
406	C₂×C₄	—	4	2	no	—
407	C₂×C₄	—	4	2	no	—
408	C₂×C₄	—	4	2	no	—
409	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
410	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
411	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
412	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
413	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
414	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
415	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
416	C₈	—	2	2	no	—
417	C₈	—	2	2	no	—
418	C₈	—	4	2	no	—
419	C₈	—	4	2	no	—
420	C₈	C₄×D₄×C₆₇	1	2	yes	—
421	C₈	C₄×D₄×C₆₇	1	2	yes	—
422	C₈	#121×C₆₇	1	2	yes	—
423	C₈	#121×C₆₇	1	2	yes	—
424	C₂²	—	2	2	no	—
425	C₂²	—	2	2	no	—
426	C₂²	—	2	4	no	—
427	C₂²	—	4	2	no	—
428	C₂²	—	4	2	no	—
429	C₂²	—	4	2	no	—
430	C₂²	—	4	4	no	—
431	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
432	C₂²	#525×C₆₇	1	1	yes	characteristic
433	C₂²	#529×C₆₇	1	1	yes	characteristic
434	C₄	—	2	2	no	—
435	C₄	—	4	2	no	—
436	C₄	—	4	2	no	—
437	C₄	—	4	2	no	—
438	C₄	—	4	4	no	—
439	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
440	C₄	C₄×#41×C₆₇	1	1	yes	characteristic
441	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
442	C₄	#532×C₆₇	1	1	yes	characteristic
443	C₂	—	2	1	no	—
444	C₂	—	2	1	no	—
445	C₂	—	4	2	no	—
446	C₂	—	8	2	no	—
447	C₂	C₄×#121×C₆₇	1	1	yes	characteristic
448	C₂	C₆₇×#2955	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₆₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₆₇×#3054	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₆₇ ◁ C₂³×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#443×C₆₇	C₂	semidirect
3. C₆₇×#2929	C₂	semidirect
4. C₂×#405×C₆₇	C₂	semidirect
5. C₆₇×#1975	C₂	semidirect
6. C₆₇×#1946	C₂	semidirect
7. C₆₇×#1938	C₂	semidirect
8. C₆₇×#1897	C₂	semidirect
9. C₆₇×#1811	C₂	semidirect
10. C₆₇×#1772	C₂	semidirect
11. C₆₇×#1764	C₂	semidirect
18. #470×C₆₇	C₄	semidirect
19. #469×C₆₇	C₄	semidirect
20. #468×C₆₇	C₄	semidirect
95. C₄×C₈×C₆₇	D₄	semidirect
218. #21548	C₆₇	direct (N × H)
226. C₂×#443	C₂×C₆₇	semidirect
227. #2929	C₂×C₆₇	semidirect
228. C₂×#405	C₂×C₆₇	semidirect
229. #1975	C₂×C₆₇	semidirect
230. #1946	C₂×C₆₇	semidirect
231. #1938	C₂×C₆₇	semidirect
232. #1897	C₂×C₆₇	semidirect
233. #1811	C₂×C₆₇	semidirect
234. #1772	C₂×C₆₇	semidirect
235. #1764	C₂×C₆₇	semidirect
236. C₆₇	#21548	direct (N × H)
243. #470	C₄×C₆₇	semidirect
244. #469	C₄×C₆₇	semidirect
245. #468	C₄×C₆₇	semidirect
320. 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
C₆₇×#3054	2
C₂×#443×C₆₇	2
C₆₇×#2929	2
C₂×#405×C₆₇	2
C₆₇×#1975	2
C₆₇×#1946	2
C₆₇×#1938	2
C₆₇×#1897	2
C₆₇×#1811	2
C₆₇×#1772	2
C₆₇×#1764	2
#21548	67

Subgroups — C67×#21548 #30700799

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21548 #30700799