Subgroups — C₆₇×#21230 #30700481

Order 17152.

1566 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#135×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#107×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#417×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2311	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1973	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1970	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1969	C₂	1	2	yes	maximal, complemented
9	C₂²×#41×C₆₇	C₂²	1	2	yes	—
10	C₂×#135×C₆₇	—	2	4	no	—
11	C₂×#135×C₆₇	—	2	4	no	—
12	C₂×#135×C₆₇	C₄	1	8	yes	complemented
13	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
15	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
20	C₂×#107×C₆₇	—	2	4	no	—
21	C₂×#107×C₆₇	C₂²	1	2	yes	—
22	C₂×#107×C₆₇	C₂²	1	2	yes	—
23	#417×C₆₇	—	2	8	no	—
24	C₂×#103×C₆₇	C₂²	1	4	yes	—
25	C₂×#102×C₆₇	C₂²	1	4	yes	—
26	#405×C₆₇	—	2	4	no	—
27	C₂×#97×C₆₇	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
29	#391×C₆₇	C₂²	1	4	yes	—
30	#390×C₆₇	C₂²	1	4	yes	—
31	#386×C₆₇	—	2	4	no	—
32	#380×C₆₇	—	2	4	no	—
33	C₂×#41×C₆₇	—	2	4	no	—
34	C₂×#41×C₆₇	—	2	4	no	—
35	C₂×#41×C₆₇	—	2	4	no	—
36	C₂×#41×C₆₇	—	2	4	no	—
37	C₂×#41×C₆₇	—	2	4	no	—
38	C₂×#41×C₆₇	C₂×C₄	1	8	yes	—
39	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
40	C₂²×D₄×C₆₇	—	2	2	no	—
41	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
42	C₂³×C₄×C₆₇	—	2	2	no	—
43	C₂³×C₄×C₆₇	—	2	2	no	—
44	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
45	#135×C₆₇	—	2	32	no	—
46	C₂×Q₁₆×C₆₇	—	2	2	no	—
47	C₂×Q₁₆×C₆₇	—	2	2	no	—
48	C₂×Q₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
49	C₂×SD₁₆×C₆₇	—	2	4	no	—
50	C₂×SD₁₆×C₆₇	—	2	4	no	—
51	C₂×SD₁₆×C₆₇	C₂×C₄	1	8	yes	complemented
52	C₂×D₈×C₆₇	—	2	2	no	—
53	C₂×D₈×C₆₇	—	2	2	no	—
54	C₂×D₈×C₆₇	C₂×C₄	1	4	yes	complemented
55	C₂×M₁₆×C₆₇	—	2	2	no	—
56	C₂×M₁₆×C₆₇	—	2	2	no	—
57	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
58	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
59	C₂²×C₈×C₆₇	D₄	1	4	yes	—
60	C₂²×C₈×C₆₇	D₄	1	4	yes	—
61	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
62	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
64	#117×C₆₇	—	2	2	no	—
65	#117×C₆₇	—	2	2	no	—
66	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
67	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
68	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
69	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
70	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
71	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
72	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
73	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
74	#107×C₆₇	—	2	4	no	—
75	#107×C₆₇	—	2	4	no	—
76	#103×C₆₇	—	2	8	no	—
77	#102×C₆₇	—	2	8	no	—
78	#97×C₆₇	—	2	4	no	—
79	C₄×C₈×C₆₇	—	2	4	no	—
80	#95×C₆₇	—	2	8	no	—
81	C₂⁴×C₆₇	—	2	2	no	—
82	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
83	#41×C₆₇	—	2	16	no	—
84	#41×C₆₇	—	2	16	no	—
85	#41×C₆₇	—	4	8	no	—
86	#41×C₆₇	—	4	8	no	—
87	C₂×Q₈×C₆₇	—	2	2	no	—
88	C₂×Q₈×C₆₇	—	2	2	no	—
89	C₂×Q₈×C₆₇	—	2	2	no	—
90	C₂×Q₈×C₆₇	—	2	2	no	—
91	C₂×Q₈×C₆₇	—	2	2	no	—
92	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
93	C₂×D₄×C₆₇	—	2	8	no	—
94	C₂×D₄×C₆₇	—	2	2	no	—
95	C₂×D₄×C₆₇	—	2	2	no	—
96	C₂×D₄×C₆₇	—	2	2	no	—
97	C₂×D₄×C₆₇	—	2	2	no	—
98	C₂×D₄×C₆₇	—	2	2	no	—
99	C₂×D₄×C₆₇	—	4	4	no	—
100	C₂×D₄×C₆₇	—	4	4	no	—
101	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
102	C₂²×C₄×C₆₇	—	2	2	no	—
103	C₂²×C₄×C₆₇	—	2	8	no	—
104	C₂²×C₄×C₆₇	—	2	4	no	—
105	C₂²×C₄×C₆₇	—	4	4	no	—
106	C₂²×C₄×C₆₇	—	4	4	no	—
107	C₂²×C₄×C₆₇	—	4	4	no	—
108	C₂²×C₄×C₆₇	—	8	4	no	—
109	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
110	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
111	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
112	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
113	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
115	C₂²×C₄×C₆₇	#41	1	2	yes	—
116	C₂²×C₄×C₆₇	#41	1	2	yes	—
117	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
118	Q₁₆×C₆₇	—	2	8	no	—
119	SD₁₆×C₆₇	—	2	16	no	—
120	D₈×C₆₇	—	2	8	no	—
121	M₁₆×C₆₇	—	2	8	no	—
122	C₂×C₈×C₆₇	—	2	2	no	—
123	C₂×C₈×C₆₇	—	2	2	no	—
124	C₂×C₈×C₆₇	C₂²⋊C₄	1	16	yes	complemented
125	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
126	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
127	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
128	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
129	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
130	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
131	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
132	C₄⋊C₄×C₆₇	—	2	4	no	—
133	C₄⋊C₄×C₆₇	—	2	4	no	—
134	C₄⋊C₄×C₆₇	—	2	4	no	—
135	C₄⋊C₄×C₆₇	—	4	4	no	—
136	C₂²⋊C₄×C₆₇	—	2	4	no	—
137	C₂²⋊C₄×C₆₇	—	8	8	no	—
138	C₄²×C₆₇	—	2	2	no	—
139	C₄²×C₆₇	—	2	2	no	—
140	C₂³×C₆₇	—	2	2	no	—
141	C₂³×C₆₇	—	2	2	no	—
142	C₂³×C₆₇	—	2	4	no	—
143	C₂³×C₆₇	—	4	2	no	—
144	C₂³×C₆₇	—	4	2	no	—
145	C₂³×C₆₇	—	4	4	no	—
146	C₂³×C₆₇	—	4	4	no	—
147	C₂³×C₆₇	C₄×D₄	1	2	yes	—
148	C₂³×C₆₇	#121	1	2	yes	—
149	C₂³×C₆₇	#127	1	2	yes	—
150	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
151	Q₈×C₆₇	—	2	4	no	—
152	Q₈×C₆₇	—	2	4	no	—
153	Q₈×C₆₇	—	4	2	no	—
154	Q₈×C₆₇	—	4	2	no	—
155	D₄×C₆₇	—	2	4	no	—
156	D₄×C₆₇	—	2	4	no	—
157	D₄×C₆₇	—	4	16	no	—
158	D₄×C₆₇	—	4	2	no	—
159	D₄×C₆₇	—	4	2	no	—
160	C₂×C₄×C₆₇	—	2	4	no	—
161	C₂×C₄×C₆₇	—	2	4	no	—
162	C₂×C₄×C₆₇	—	2	4	no	—
163	C₂×C₄×C₆₇	—	2	4	no	—
164	C₂×C₄×C₆₇	—	4	2	no	—
165	C₂×C₄×C₆₇	—	4	2	no	—
166	C₂×C₄×C₆₇	—	4	16	no	—
167	C₂×C₄×C₆₇	—	4	4	no	—
168	C₂×C₄×C₆₇	—	4	8	no	—
169	C₂×C₄×C₆₇	—	8	8	no	—
170	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
171	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
172	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
173	C₂×C₄×C₆₇	#121	1	4	yes	—
174	C₂×C₄×C₆₇	#121	1	2	yes	—
175	C₂×C₄×C₆₇	#124	1	4	yes	—
176	C₂×C₄×C₆₇	#127	1	2	yes	—
177	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
178	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
179	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
180	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
181	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
182	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
183	C₈×C₆₇	—	2	4	no	—
184	C₈×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
185	C₈×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
186	C₂²×C₆₇	—	2	2	no	—
187	C₂²×C₆₇	—	2	2	no	—
188	C₂²×C₆₇	—	2	4	no	—
189	C₂²×C₆₇	—	2	4	no	—
190	C₂²×C₆₇	—	4	8	no	—
191	C₂²×C₆₇	—	4	4	no	—
192	C₂²×C₆₇	—	8	2	no	—
193	C₂²×C₆₇	—	8	2	no	—
194	C₂²×C₆₇	—	8	4	no	—
195	C₂²×C₆₇	#390	1	4	yes	—
196	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
197	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
198	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
199	C₂²×C₆₇	#576	1	2	yes	—
200	C₂²×C₆₇	#578	1	2	yes	—
201	C₄×C₆₇	—	4	4	no	—
202	C₄×C₆₇	—	4	4	no	—
203	C₄×C₆₇	—	8	4	no	—
204	C₄×C₆₇	#390	1	4	yes	—
205	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
206	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
207	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
208	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
209	#21230	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
210	C₂×C₆₇	—	2	4	no	—
211	C₂×C₆₇	—	8	4	no	—
212	C₂×C₆₇	C₂×#390	1	1	yes	characteristic
213	C₂×C₆₇	#2951	1	2	yes	—
214	C₂×C₆₇	#3061	1	2	yes	—
215	C₂×C₆₇	#3155	1	1	yes	characteristic
216	C₂×C₆₇	#3156	1	1	yes	characteristic
217	C₂²×#135	C₂×C₆₇	1	1	yes	characteristic
218	C₂²×#107	C₂×C₆₇	1	1	yes	characteristic, complemented
219	C₂×#417	C₂×C₆₇	1	2	yes	complemented
220	C₂×#405	C₂×C₆₇	1	1	yes	characteristic, complemented
221	#2311	C₂×C₆₇	1	2	yes	complemented
222	#1973	C₂×C₆₇	1	4	yes	complemented
223	#1970	C₂×C₆₇	1	2	yes	complemented
224	#1969	C₂×C₆₇	1	2	yes	complemented
225	C₆₇	#21230	1	1	yes	characteristic, direct product factor, Sylow
226	C₂²×#41	C₂²×C₆₇	1	2	yes	—
227	C₂×#135	—	2	4	no	—
228	C₂×#135	—	2	4	no	—
229	C₂×#135	C₄×C₆₇	1	8	yes	complemented
230	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
231	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
232	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
233	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
234	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
235	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
236	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
237	C₂×#107	—	2	4	no	—
238	C₂×#107	C₂²×C₆₇	1	2	yes	—
239	C₂×#107	C₂²×C₆₇	1	2	yes	—
240	#417	—	2	8	no	—
241	C₂×#103	C₂²×C₆₇	1	4	yes	—
242	C₂×#102	C₂²×C₆₇	1	4	yes	—
243	#405	—	2	4	no	—
244	C₂×#97	C₂²×C₆₇	1	2	yes	—
245	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
246	#391	C₂²×C₆₇	1	4	yes	—
247	#390	C₂²×C₆₇	1	4	yes	—
248	#386	—	2	4	no	—
249	#380	—	2	4	no	—
250	C₂×#41	—	2	4	no	—
251	C₂×#41	—	2	4	no	—
252	C₂×#41	—	2	4	no	—
253	C₂×#41	—	2	4	no	—
254	C₂×#41	—	2	4	no	—
255	C₂×#41	C₂×C₄×C₆₇	1	8	yes	—
256	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
257	C₂²×D₄	—	2	2	no	—
258	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
259	C₂³×C₄	—	2	2	no	—
260	C₂³×C₄	—	2	2	no	—
261	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
262	#135	—	2	32	no	—
263	C₂×Q₁₆	—	2	2	no	—
264	C₂×Q₁₆	—	2	2	no	—
265	C₂×Q₁₆	C₂×C₄×C₆₇	1	4	yes	complemented
266	C₂×SD₁₆	—	2	4	no	—
267	C₂×SD₁₆	—	2	4	no	—
268	C₂×SD₁₆	C₂×C₄×C₆₇	1	8	yes	complemented
269	C₂×D₈	—	2	2	no	—
270	C₂×D₈	—	2	2	no	—
271	C₂×D₈	C₂×C₄×C₆₇	1	4	yes	complemented
272	C₂×M₁₆	—	2	2	no	—
273	C₂×M₁₆	—	2	2	no	—
274	C₂×M₁₆	D₄×C₆₇	1	4	yes	—
275	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
276	C₂²×C₈	D₄×C₆₇	1	4	yes	—
277	C₂²×C₈	D₄×C₆₇	1	4	yes	—
278	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
279	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
280	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
281	#117	—	2	2	no	—
282	#117	—	2	2	no	—
283	C₂×C₄⋊C₄	—	2	4	no	—
284	C₂×C₄⋊C₄	—	2	4	no	—
285	C₂×C₄⋊C₄	—	2	2	no	—
286	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
287	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
288	C₂×C₂²⋊C₄	—	2	8	no	—
289	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
290	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
291	#107	—	2	4	no	—
292	#107	—	2	4	no	—
293	#103	—	2	8	no	—
294	#102	—	2	8	no	—
295	#97	—	2	4	no	—
296	C₄×C₈	—	2	4	no	—
297	#95	—	2	8	no	—
298	C₂⁴	—	2	2	no	—
299	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
300	#41	—	2	16	no	—
301	#41	—	2	16	no	—
302	#41	—	4	8	no	—
303	#41	—	4	8	no	—
304	C₂×Q₈	—	2	2	no	—
305	C₂×Q₈	—	2	2	no	—
306	C₂×Q₈	—	2	2	no	—
307	C₂×Q₈	—	2	2	no	—
308	C₂×Q₈	—	2	2	no	—
309	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
310	C₂×D₄	—	2	8	no	—
311	C₂×D₄	—	2	2	no	—
312	C₂×D₄	—	2	2	no	—
313	C₂×D₄	—	2	2	no	—
314	C₂×D₄	—	2	2	no	—
315	C₂×D₄	—	2	2	no	—
316	C₂×D₄	—	4	4	no	—
317	C₂×D₄	—	4	4	no	—
318	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
319	C₂²×C₄	—	2	2	no	—
320	C₂²×C₄	—	2	8	no	—
321	C₂²×C₄	—	2	4	no	—
322	C₂²×C₄	—	4	4	no	—
323	C₂²×C₄	—	4	4	no	—
324	C₂²×C₄	—	4	4	no	—
325	C₂²×C₄	—	8	4	no	—
326	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
327	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
328	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
329	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
330	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
331	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
332	C₂²×C₄	#41×C₆₇	1	2	yes	—
333	C₂²×C₄	#41×C₆₇	1	2	yes	—
334	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
335	Q₁₆	—	2	8	no	—
336	SD₁₆	—	2	16	no	—
337	D₈	—	2	8	no	—
338	M₁₆	—	2	8	no	—
339	C₂×C₈	—	2	2	no	—
340	C₂×C₈	—	2	2	no	—
341	C₂×C₈	C₂²⋊C₄×C₆₇	1	16	yes	complemented
342	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
343	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
344	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
345	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
346	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
347	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
348	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
349	C₄⋊C₄	—	2	4	no	—
350	C₄⋊C₄	—	2	4	no	—
351	C₄⋊C₄	—	2	4	no	—
352	C₄⋊C₄	—	4	4	no	—
353	C₂²⋊C₄	—	2	4	no	—
354	C₂²⋊C₄	—	8	8	no	—
355	C₄²	—	2	2	no	—
356	C₄²	—	2	2	no	—
357	C₂³	—	2	2	no	—
358	C₂³	—	2	2	no	—
359	C₂³	—	2	4	no	—
360	C₂³	—	4	2	no	—
361	C₂³	—	4	2	no	—
362	C₂³	—	4	4	no	—
363	C₂³	—	4	4	no	—
364	C₂³	C₄×D₄×C₆₇	1	2	yes	—
365	C₂³	#121×C₆₇	1	2	yes	—
366	C₂³	#127×C₆₇	1	2	yes	—
367	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
368	Q₈	—	2	4	no	—
369	Q₈	—	2	4	no	—
370	Q₈	—	4	2	no	—
371	Q₈	—	4	2	no	—
372	D₄	—	2	4	no	—
373	D₄	—	2	4	no	—
374	D₄	—	4	16	no	—
375	D₄	—	4	2	no	—
376	D₄	—	4	2	no	—
377	C₂×C₄	—	2	4	no	—
378	C₂×C₄	—	2	4	no	—
379	C₂×C₄	—	2	4	no	—
380	C₂×C₄	—	2	4	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	4	16	no	—
384	C₂×C₄	—	4	4	no	—
385	C₂×C₄	—	4	8	no	—
386	C₂×C₄	—	8	8	no	—
387	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
388	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
389	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
390	C₂×C₄	#121×C₆₇	1	4	yes	—
391	C₂×C₄	#121×C₆₇	1	2	yes	—
392	C₂×C₄	#124×C₆₇	1	4	yes	—
393	C₂×C₄	#127×C₆₇	1	2	yes	—
394	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
395	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
396	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
397	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
398	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
399	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
400	C₈	—	2	4	no	—
401	C₈	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
402	C₈	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
403	C₂²	—	2	4	no	—
404	C₂²	—	2	2	no	—
405	C₂²	—	2	2	no	—
406	C₂²	—	2	4	no	—
407	C₂²	—	4	4	no	—
408	C₂²	—	4	8	no	—
409	C₂²	—	8	4	no	—
410	C₂²	—	8	2	no	—
411	C₂²	—	8	2	no	—
412	C₂²	#390×C₆₇	1	4	yes	—
413	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
414	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
415	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
416	C₂²	#576×C₆₇	1	2	yes	—
417	C₂²	#578×C₆₇	1	2	yes	—
418	C₄	—	4	4	no	—
419	C₄	—	4	4	no	—
420	C₄	—	8	4	no	—
421	C₄	#390×C₆₇	1	4	yes	—
422	C₄	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
423	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
424	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
425	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
426	C₂	—	2	4	no	—
427	C₂	—	8	4	no	—
428	C₂	C₂×#390×C₆₇	1	1	yes	characteristic
429	C₂	C₆₇×#2951	1	2	yes	—
430	C₂	C₆₇×#3061	1	2	yes	—
431	C₂	C₆₇×#3155	1	1	yes	characteristic
432	C₂	C₆₇×#3156	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×#135×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×#107×C₆₇	C₂	semidirect
3. C₂×#417×C₆₇	C₂	semidirect
4. C₂×#405×C₆₇	C₂	semidirect
5. C₆₇×#2311	C₂	semidirect
6. C₆₇×#1973	C₂	semidirect
7. C₆₇×#1970	C₂	semidirect
8. C₆₇×#1969	C₂	semidirect
12. C₂×#135×C₆₇	C₄	semidirect
48. C₂×Q₁₆×C₆₇	C₂×C₄	semidirect
51. C₂×SD₁₆×C₆₇	C₂×C₄	semidirect
54. C₂×D₈×C₆₇	C₂×C₄	semidirect
124. C₂×C₈×C₆₇	C₂²⋊C₄	semidirect
209. #21230	C₆₇	direct (N × H)
218. C₂²×#107	C₂×C₆₇	semidirect
219. C₂×#417	C₂×C₆₇	semidirect
220. C₂×#405	C₂×C₆₇	semidirect
221. #2311	C₂×C₆₇	semidirect
222. #1973	C₂×C₆₇	semidirect
223. #1970	C₂×C₆₇	semidirect
224. #1969	C₂×C₆₇	semidirect
225. C₆₇	#21230	direct (N × H)
229. C₂×#135	C₄×C₆₇	semidirect
265. C₂×Q₁₆	C₂×C₄×C₆₇	semidirect
268. C₂×SD₁₆	C₂×C₄×C₆₇	semidirect
271. C₂×D₈	C₂×C₄×C₆₇	semidirect
341. C₂×C₈	C₂²⋊C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#135×C₆₇	2
C₂²×#107×C₆₇	2
C₂×#417×C₆₇	2
C₂×#405×C₆₇	2
C₆₇×#2311	2
C₆₇×#1973	2
C₆₇×#1970	2
C₆₇×#1969	2
#21230	67

Subgroups — C67×#21230 #30700481

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21230 #30700481